THE   REALITIES   OF   MODERN   SCIENCE 


THE  MACMILLAN  COMPANY 

NEW  YORK    •    BOSTON   •    CHICAGO  •    DALLAS 
ATLANTA   •    SAN    FRANCISCO 

MACMILLAN  &   CO.,  LIMITED 

LONDON  •    BOMBAY  •    CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  LTD. 

TORONTO 


THE  REALITIES  OF  MODERN 
SCIENCE 

AN  INTRODUCTION  FOR  THE 
GENERAL  READER 


BY 


JOHN  MILLS 

RESEARCH    LABORATORIES,    WESTERN   ELECTRIC    COMPANY,   INC.; 

MEMBER,    AMERICAN    PHYSICAL   SOCIETY  ;    AUTHOR   OF 

44  RADIO   COMMUNICATION,"    KTC. 


Ntfo 

THE  MACMILLAN  COMPANY 
1919 

All  rights  referred 


COPYRIGHT,  1919, 
BY  THE  MACMILLAN  COMPANY. 


Set  up  and  electrotyped.    Published  May,  1919. 


XortoooB 

J.  S.  Cushing  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


TO 


397609 


PREFACE 

THE  present  volume  is  intended  for  the  general 
reader,  interested  in  modern  science,  who  finds  few 
clews  to  recent  advances  in  his  memories  of  the  formal 
instruction  of  school  or  college  days. 

During  the  last  twenty  years  physical  research  has 
penetrated  the  mysteries  of  the  chemical  elements  and 
has  demonstrated  that  their  atoms  are  granular  in 
structure  and  electrical  in  nature.  Transmutation 
-  the  dream  of  the  alchemists  —  is  to-day  recognized 
as  a  natural  process  in  the  case  of  radioactive  sub- 
stances. An  element  common  to  all  matter  has  been 
found  in  the  electron  which  has  been  isolated  and 
studied.  It  is  a  reality  of  modern  science  and  in 
terms  of  it  scientists  are  rapidly  explaining  the  phe- 
nomena of  all  physical  science,  the  essential  unity  of 
which  its  discovery  has  emphasized. 

The  existence  of  electrons  and  their  determining 
effect  in  the  composition  of  the  chemical  elements  are 
easily  demonstrable  facts,  compared  to  which  the  in- 
destructibility of  matter  is  a  speculative  assertion  and 
the  independence  of  mass  and  speed  an  exploded  theory. 
Nevertheless  our  education  is  so  constrained  by 
established  curricula  and  text-books  adapted  thereto, 
that  a  knowledge  of  these  facts  is  usually  acquired 
only  by  college  students  who  elect  the  more  advanced 
courses.  Even  in  the  subject  of  electricity  most  of 

vii 


viii  PREFACE 

the  phenomena  are  treated  in  elementary  courses 
without  reference  to  the  electron,  the  fundamental 
entity. 

One  difficulty  seems  to  be  that  physical  science  is 
subdivided,  classified,  and  named  until  the  unity  is 
completely  obscured.  Of  each  subdivision  the  accepted 
method  of  instruction  assigns  certain  phenomena  to 
high  school  or  sophomore  courses  and  others  to  junior 
work.  The  results  of  recent  researches  are  postponed 
for  consideration  in  graduate  courses  and  " recent" 
often  means  since  the  graduation  of  the  instructor. 
The  selection  of  material  for  elementary  courses  is  not 
always  determined  by  whether  or  not  the  phenomena 
are  fundamental  and  easily  comprehensible,  for  it  is 
largely  traditional.  For  example,  the  writer  remembers 
discussing  a  well-known  text-book  with  a  high  school 
teacher,  who  said  it  was  not  adapted  to  his  field  be- 
cause it  introduced  the  "Brownian  movement."  The 
writer,  however,  believes  that  the  Brownian  movement 
is  capable  of  simple  presentation  (cf.  pp.  155  and  158) 
and  should  be  discussed  in  order  that  the  student  may 
obtain  satisfactory  concepts  upon  which  to  base  his 
future  study. 

So  far  as  concerns  college  instruction  one  alternative 
is  a  general  introductory  course  which  cuts  across  the 
conventional  divisions  of  subject  matter,  selecting  to 
emphasize  the  unity.  Nor  should  this  introduction 
have  the  finality  of  the  usual  elementary  course,  for  it 
should  indicate  the  evolutionary  character  of  science 
and  lead  up  to  present  problems.  It  should  also  indi- 
cate the  historical  aspects  and  the  social  significance 
of  modern  science.  In  material  it  should  be  adapted 


PREFACE  ix 

to  the  future  citizen  rather  than  the  future  scientist, 
to  general  readers  rather  than  technical  students.  The 
present  book  follows  one  of  the  many  possible  outlines 
for  such  an  introduction  and  constitutes  a  suggestion 
as  to  one  of  the  changes  in  teaching  methods  which 
our  educational  system  seems  to  require. 

The  general  reader  is  under  no  compulsion  from  a 
traditional  curriculum  and  may  pick  and  choose  his 
sources  of  information.  To  the  study  of  science  he 
may,  however,  need  an  introduction  and  this  need  the 
present  volume  attempts  to  satisfy.  The  earlier 
chapters  following  an  historical  order  develop  without 
abstract  formulations  the  fundamental  concepts  of 
modern  science.  By  the  time  Chapter  IX  is  reached 
the  necessity  of  algebraic  expression  has  become 
apparent  and  the  three  succeeding  chapters  illustrate 
its  usefulness  while  developing  the  ideas  of  defining 
equations,  rates,  and  the  laws  of  motion.  Not  all 
these  ideas  are  symbolized  and  the  reader  who  feels 
an  ineptitude  for  mathematical  expressions  is  advised 
to  give  these  chapters  a  cursory  and  mechanical  reading 
in  order  to  reach  the  important  chapters  on  molecular 
and  electronic  motions. 

The  majority  of  readers  will  probably  find  no 
obstacles  in  the  occasional  formulas  of  the  later  text. 
For  these  the  book  may  serve  as  an  introduction  to  the 
general  literature  of  physical  science.  To  this  end 
footnotes  refer  to  books  on  special  subjects,  the  greater 
portions  of  each  of  which  the  reader  will  find  compre- 
hensible in  case  he  is  interested  in  more  detailed  study. 

The  writer  wishes  to  make  an  acknowledgment  to 
the  Physics  Department  of  Mount  Holyoke  College, 


X  PREFACE 

Professor  Laird,  the  head  of  the  Department,  and 
Dr.  Shields  read  this  book  in  manuscript  during 
the  summer  of  1918.  Miss  Shields  has  followed  the 
general  method  and  incorporated  some  of  the  ma- 
terial in  a  course  for  second  year  students.  For  this 
proximate  use  of  the  manuscript  and  for  the  sugges- 
tions which  have  arisen  from  it  the  writer  expresses 
his  thanks  to  Miss  Shields.  Thanks  are  also  expressed 
to  Dr.  K.  K.  Darrow  of  the  Western  Electric  Company, 
who  has  read  the  proof. 

J.  M. 

RESEARCH  LABORATORIES 
THE  WESTERN  ELECTRIC  COMPANY,  INC. 
March  1,  1919 


CONTENTS 

CHAPTER  PAOE 

PREFACE vii 

I  THE  BEGINNINGS  OF  KNOWLEDGE     ....  1 

II  THE  MACHINES  OF  THE  ANCIENTS    ....  14 

III  WEIGHTS  AND  MEASURES 24 

IV  THE  BEGINNINGS  OF  SCIENCE 35 

V  THE  BEGINNINGS  OF  EXPERIMENTATION    ...  48 

VI  THE  REALITIES  OF  SCIENCE 60 

VII  THE  MOLECULAR  COMPOSITION  OF  MATTER     .        .  70 

VIII  THE  ELECTRON 88 

IX  ENERGY 101 

X  SOME  USES  OF  MATHEMATICS 115 

XI  RATES 128 

XII  FORCE,  A  SPACE  RATE  OF  ENERGY           .        .        .  139 

XIII  MOLECULAR  MOTIONS  AND  TEMPERATURE        .        .  155 

XIV  MOTIONS  OF  ELECTRONS 173 

XV  THE  INTERACTIONS  OF  MOVING  ELECTRONS      .        .  195 

XVI  THE  CONTINUITY  AND  CORRESPONDENCE  OF  MOLEC- 
ULAR STATES 217 

XVII  MOLECULAR  MIXTURES 234 

XVIII  ELECTROLYTIC  DISSOCIATION 244 

XIX  EQUILIBRIA  AND  THEIR  DISPLACEMENT     .        .        .  257 

XX  MOLECULAR  MAGNITUDES 270 

XXI  MOLECULAR  ENERGY 288 

XXII  ELECTRONIC  MAGNITUDES 304 

INDEX 323 

xi 


THE   REALITIES   OF   MODERN 
SCIENCE 

CHAPTER  I 
THE  BEGINNINGS  OF  KNOWLEDGE 

MAN  is  sometimes  distinguished  as  a  "tool  using" 
animal.  Monkeys  may  be  taught  a  few  simple  opera- 
tions with  tools,  such  as  cracking  nuts  with  a  stone,  but 
usually  they  merely  mimic  a  man.  Their  hands,  with 
five  fingers  instead  of  four  and  a  thumb,  are  not  suitable 
for  picking  up  objects.  Other  animals  have  bodies  the 
parts  of  which  are  convenient  for  various  purposes,  as 
horns  for  defense  or  claws  for  offense,  but  parts  of  the 
body,  even  though  they  serve  a  special  purpose,  are 
not  tools.  A  tool  is  an  object  apart  from  its  user. 

Man's  superiority  as  a  tool  user,  partly  physical  in 
his  ability  to  stand  erect  and  use  his  hands,  and  partly 
mental,  ultimately  secured  his  position  over  the  beasts 
of  the  world  and  enabled  him  to  construct  our  material 
civilization.  An  instance  of  the  mental  difference  is 
found  in  the  attitude  toward  fire.  The  dog  knows 
the  pleasure  on  a  cold  day  of  curling  up  beside  the  fire. 
For  thousands  of  years  he  has  been  man's  companion 
and  intelligent  servant.  Nevertheless,  a  dog,  shivering 
before  a  dying  fire,  lacks  the  ability  to  foresee  effects, 


2  TKU  REAL 'TIES  OF  MODERN  SCIENCE 

which  would  lead  him  to  push  a  new  stick  into  the 
smoldering  embers. 

No  one  knows  how  man  first  learned  to  keep  fire 
alive  and  how  ages  later  he  learned  to  make  a  fire. 
Many  students  of  the  relics  of  man  in  earlier  geologic 
ages  think  that  he  first  saw  fire  in  the  dry  seasons 
when  lightning  had  started  it.  Perhaps  after  a  brush 
fire  had  been  checked  by  a  rain,  some  man,  more  adven- 
turous and  curious  than  his  fellows,  approached  and 
picked  up  a  burning  stick.  The  man  who  was  the 
first  in  the  world  to  pick  up  a  brand  was  not  only 
brave  but  an  important  discoverer,  in  showing  that 
fire  could  be  safely  handled.  Nobody  told  him  that 
although  wood  is  combustible  and  if  heated  to  the 
proper  temperature  will  ignite  with  air  and  burn,  still, 
when  below  this  temperature,  it  is  a  very  good  heat 
insulator.  The  experimental  fact  preceded  by  ages 
its  scientific  description. 

Probably  other  members  of  this  discoverer's  family 
imitated  him,  just  as  small  boys  around  their  first 
bonfire  will  soon  follow  the  lead  of  the  older  boy  who 
brandishes  a  flaming  stick.  From  this  time  familiarity 
with  fire  increased,  although  the  first  fire-handlers 
may  have  died  before  they  again  saw  fire,  leaving  the 
discovery  to  be  remade  by  a  later  generation. 

Perhaps,  chilled  by  the  rain  which  checked  the 
spreading  bush  fire,  our  man  may  have  discovered 
the  pleasantness  of  warming  his  own  body  beside  the 
embers.  He  may  also  have  tasted  roasted  meat  for 
the  first  time,  driven  by  hunger  to  try  the  half  burned 
body  of  some  small  animal  which  was  caught  in  the 
fire.  If  so,  it  was  a  long  time  before  he  ate  cooked 


THE  BEGINNINGS  OF  KNOWLEDGE  3 

meat  again,  for  fires  could  only  start  in  the  dry  season 
from  lightning  or  from  a  volcanic  eruption. 

In  the  next  dry  season  our  man  may  have  remembered 
his  first  taste  of  cooked  meat.  More  probably  he  did 
not  remember  at  all.  Certainly  his  brain  did  not 
permit  of  his  looking  ahead  and  saying,  "Next  dry 
season,  if  there  is  a  fire,  I  may  find  some  roasted  meat." 
His  mind  developed  slowly,  and  the  first  experiment 
had  to  be  performed  many  times  by  different  members 
of  the  race  before  he  was  ready  for  his  second  step  in 
the  domestication  of  fire.  History1  incompletely  de- 
picts these  slow  ages,  the  dark  nights,  the  cold  and 
rainy  days,  and  the  uncooked  food. 

How  man  learned  to  keep  a  fire  alive  is  also  a  matter 
of  surmise.  Perhaps  he  noted  that  it  lasted  longest 
in  large  heaps  of  damp  brush  or  in  the  trunks  of  rotten 
trees.  At  some  such  remains  of  a  fire  given  by  lightning 
from  the  heavens  he  may  have  cooked  meats  and 
noticed  that  the  fire  flared  up  under  the  breeze.  An- 
other discoverer  may  have  carried  home  to  his  cave 
a  glowing  brand,  which  when  thrown  aside  into  the 
dry  branches  started  a  larger  fire.  By  such  a  series 
of  accidents  the  discovery  may  have  been  made  that 
fire  could  be  kept  by  furnishing  it  with  fuel,  and  that 
it  burned  less  fiercely  and  lasted  longer  if  covered  from 
the  wind. 

As  yet  no  way  was  known  of  kindling  a  fire.  If  the 
fire  died,  the  tribe  had  to  beg  or  steal  from  another 
tribe  or  wait  until  a  dry  season  when  nature  again 
provided  fire.  To-day  we  little  realize  the  situation  of 

1  An  interesting  sketch  of  this  evolution  is  that  of  Migeod, 
"  Earliest  Man,"  Kegan  Paul,  London,  1916. 


4  THE  REALITIES  OF  MODERN  SCIENCE 

mankind  when  its  scientific  knowledge  was  only  suf- 
ficient to  maintain  a  fire  and  not  to  start  one.  It 
is  no  wonder  that  religious  rites  grew  up  about  the 
tribal  fire  and  that  special  caretakers  were  dedicated 
to  it. 

While  early  man  was  learning  some  of  the  uses  of  fire 
and  how  to  conserve  and  control  it,  he  was  also  develop- 
ing tools  and  through  their  use  making  advances  in 
clothing  and  shelter.  His  first  tool  might  have  been 
a  stone.  Perhaps,  while  lying  on  the  floor  of  his  cave, 
he  was  frightened  by  the  entrance  of  a  wild  beast  and 
his  hand  closed  over  one  of  the  small  stones  which  he 
had  earlier  brushed  aside  to  make  a  smoother  bed.  He 
fought  blindly  and  the  stone  slipped  from  his  grasp. 
A  chance  hit,  the  astonished  beast  turned  tail,  and  he 
was  saved. 

The  next  time  he  was  attacked  he  may  have  remem- 
bered the  episode  of  the  cave  and  used  a  stone  again. 
If  his  brain  was  not  sufficiently  developed  for  him  to 
connect  the  act  with  the  flight  of  the  beast  he  was 
probably  eaten  for  his  ignorance.  Those  families  or 
tribes  which  quickest  learned  this  use  of  stones  were 
fittest  to  survive. 

A  peaceful  use  of  stones  as  agricultural  tools  may 
or  may  not  have  preceded  their  use  as  weapons.  Early 
man,  we  believe,  dug  for  roots  and  in  certain  seasons 
these  were  the  only  succulent  food  which  he  could 
obtain.  While  burrowing  with  his  hands  he  might 
strike  a  stone  and  instinctively  grasping  it  continue 
the  same  digging  motion.  If  the  stone  happened  to 
be  the  right  shape  he  found  to  his  surprise  that  his 
work  went  faster  and  easier.  Later  he  may  have  tried 


THE  BEGINNINGS  OF  KNOWLEDGE  5 

it  again  and  found  by  bitter  experience  that  round 
smooth  stones  retarded  his  operations.  Still  later 
some  man  may  have  broken  a  stone  by  throwing  it 
against  a  rock,  hoping  to  find  among  the  pieces  one 
suited  to  his  needs. 

In  some  such  way  there  developed  the  idea  of  using 
tools  and  of  forming  them.  Of  course,  it  is  only  a 
small  step  from  breaking  the  stone  as  just  described 
and  chipping  off  edges  of  the  selected  stone  so  as  to 
form  it  more  nearly  to  the  desired  shape.  But  this 
step  may  not  have  been  taken  for  centuries.  Even 
if  some  man  did  make  this  advance  and  his  tribe  learned 
it  from  him,  accident  or  disease  could  have  destroyed 
them  all  and  with  them  the  precious  knowledge.  There 
were  as  yet  no  records  of  past  achievements  to  save  a 
student  the  mistakes  and  trials  through  which  ages 
of  mankind  accumulated  and  arranged  its  knowledge 
of  the  physical  world. 

If  food  became  scarce  in  the  region  where  lived  a  tribal 
group  which  had  learned  to  preserve  fire  it  may  have 
wandered  into  other  regions  and  so  spread  its  special 
knowledge.  This  change  from  the  old  conditions  and 
the  necessity  of  meeting  new  ones  would  have  stimu- 
lated it  to  new  discoveries  along  other  lines,  as  perhaps 
in  the  making  of  vessels  for  carrying  fire  and  water  on 
its  travels.  In  this  migration  it  may  have  met  another 
tribe  with  other  knowledge.  If  they  fought  the  weaker 
tribe  perished  with  its  knowledge,  or  if  they  mingled 
the  enlarged  group  enjoyed  the  knowledge  common  to 
its  parts. 

Except  when  tribes  amalgamated  the  exchange  of 
information  must  have  been  very  slow  because  of  the 


6  THE  REALITIES  OF  MODERN  SCIENCE 

difficulties  of  language.  At  this  time  in  the  develop- 
ment of  man  he  was  just  learning  to  talk.  His  few 
ideas  and  fewer  words  were  concerned  with  concrete 
objects  such  as  water,  fire,  food,  sun,  and  moon  or  with 
his  own  sensations  such  as  cold,  hunger,  and  fatigue. 
He  had  no  vocabulary  for  expressing  abstract  ideas. 
Imagine  the  difficulty  of  trying  to  tell  a  primitive  man 
of  the  speed  record  of  an  automobile,  of  the  efficiency 
of  its  engine,  or  of  the  refinement  of  its  lines  and  up- 
holstery. 

Between  the  discovery  of  the  method  of  conserving 
fire  and  its  production  by  a  simple  machine  many 
thousands  of  years  must  have  intervened.  A  primitive 
method  is  that  of  rubbing  a  sharp  stick  back  and  forth 
in  a  groove  cut  in  a  block  of  hard  wood.  For  this 
concrete  case  he  had  learned  that  his  work  in  rubbing 
together  two  surfaces  resulted  in  heat.  The  heat 
produced  by  his  efforts  did  not  diffuse  rapidly  and  so 
make  the  entire  block  of  wood  a  little  warmer  than  it 
was  before.  Instead  it  was  concentrated  in  the  wood 
adjacent  to  the  groove.  If  he  rubbed  hard  enough, 
that  is,  if  he  did  enough  work,  this  part  of  the  block 
would  then  rise  rapidly  in  temperature,  smoke,  and 
finally  set  fire  to  dry  wood  dust  or  other  trader  placed 
in  the  groove. 

This  simple  device  is  not,  however,  a  machine.  It 
is  merely  a  special  tool  for  rubbing  and  a  block  to  be 
rubbed.  When  primitive  men  arranged  a  combination 
where  the  rubbing  tool  was  actuated  by  another  part 
of  the  device,  which  part  in  turn  was  controlled  by  the 
operator,  then  he  had  a  machine.  In  Fig.  1  is  shown 
such  a  fire-making  machine.  It  is  of  course  a  develop- 


THE  BEGINNINGS  OF  KNOWLEDGE 


FIG.  1. 


ment  of  an  earlier  form,  shown  in  Fig.  2,  where  a  pointed 

stick  or  drill  is  twisted  by  the  operator's  hands. 
Let   us   examine   this 

primitive  machine  more 

closely.     The   operator, 

forcing  the  bow  back  and 

forth,  causes  its  various 

points    to    move    along 

parallel  paths.    There  is 

no  turning  of  the  bow, 

so  its  motion  is  "  trans- 
lation."     The   machine 

then  converts  a  motion 

of  translation  into  one  of 

rotation.      The  cord  of 

the  bow  merely  replaces 

the  hands  of  the  operator.     In  turning  the  drill  of 

Fig.   2   through  one  complete  revolution  each  hand 

comes  in  contact  with  every 
point  of  its  circumf  erence  and 
moves  a  distance  equal  to 
the  circumf  erence.  When  the 
drill  is  operated  by  the  bow, 
the  latter  must  move  through 
an  equal  distance.  The  op- 
erator may  use  only  one  hand 
with  the  bow,  but,  obviously, 
he  must  push  or  pull  twice  as 
hard  since  his  other  hand  is 
not  assisting.  The  advantage 

of  the  machine  is  not  that  it  requires  less  work,  but 

that  it  is  more  convenient,  requiring  only  the  simple 


FIG.  2. 


8  THE  REALITIES  OF  MODERN  SCIENCE 

motion  of  pushing  and  pulling.  To  operate  the  hand 
drill  requires  skill  and  practice,  but  to  operate  the 
machine  requires  less  skill  and  the  ability  to  work. 

From  this  primitive  design  we  can  obtain  a  concept 
of  a  machine  as  a  device  in  which  a  motion  communi- 
cated to  one  part  results  in  a  different  motion  of  a 
second  part.  Now,  although  this  is  a  correct  statement, 
it  is  one  that  is  lacking  in  human  interest  or  significance. 
Early  man  devised  machines  in  his  struggle  against 
the  rigors  of  the  world  about  him.  From  our  stand- 
point, as  well  as  from  his,  we  should  consider  the  ma- 
chine with  reference  to  its  effect  on  mankind. 

It  is  work  to  draw  the  bow  back  and  forth  against 
the  friction  of  the  drill  and  the  block.  We  call  the 
ability  to  do  work  " energy"  and  say  that  when  work 
is  done  energy  is  expended.  The  machine  requires 
energy  from  some  source,  and  material  upon  which  to 
operate.  Push  and  pull,  push  and  pull,  is  all  that  it 
requires,  for  in  its  sequence  of  motions  it  accomplishes 
the  results  of  the  most  skilled  craftsman.  And  this 
sequence  of  events  is  determined  once  and  for  all  by 
the  inventor  of  the  machine.  The  needle  of  the  sewing 
machine  rises  and  falls  on  the  material  beneath  it 
whether  it  be  cloth  or  the  hand  of  the  operator.  The 
loaded  gun  shoots  friend  and  foe  alike  provided  only 
the  energy  in  its  cartridge  is  released. 

A  machine,  then,  is  a  device  for  producing  a  definite 
sequence  of  motions  provided  the  supply  of  energy 
necessary  to  operate  it  is  released.  Unless  the  ma- 
chine is  broken,  in  which  case,  of  course,  it  is  not  a 
machine  at  all,  man's  only  control  of  it  is  a  permissive 
one,  of  allowing  it  to  run  or  not,  by  controlling  its 


THE  BEGINNINGS  OF  KNOWLEDGE  9 

supply  of  energy.  In  our  modern  life  where  machines 
for  varied  purposes  are  available  this  control  is  obtained 
by  means  of  a  switch  or  button,  a  faucet  or  a  throttle. 
It  is  ours  to  say  whether  or  not  we  shall  release  the 
energy,  but  otherwise  the  operation  is  inherent  hi  the 
design  of  the  machine  except  in  so  far  as  a  counter- 
acting machine,  like  a  brake,  may  be  available.  In 
such  a  case,  this  too  is  controlled  by  its  switch,  throttle, 
or  lever. 

A  machine,  once  started  and  supplied  with  energy, 
performs  its  characteristic  series  of  operations  without 
reference  to  their  effects.  In  the  inevitability  of  the 
sequence  for  which  the  machine  is  designed  inhere  its 
advantages  and  dangers.  It  is  therefore  vitally  neces- 
sary that  those  who  control  machines  or  come  within 
the  range  of  their  motions  should  bear  hi  mind  this 
fact.  For  this  reason  the  old  adage  of  "Look  before 
you  leap  "  has  been  replaced  by  the  modern  slogan  of 
"  Safety  first." 

The  machine  requires  a  source  of  energy,  but  does 
not  demand  an  intelligence  or  skill  on  the  part  of  the 
operator  proportional  to  the  results  which  it  produces. 
While  this  permits  the  unskilled  to  produce  the  results 
of  the  skilled  it  does  not  offer  the  changing  conditions 
which  stimulate  mental  growth.  There  is  a  sameness 
to  its  operations  which  may  make  the  human  agents 
mere  appendages  of  the  machines  they  seem  to  con- 
trol. The  social  and  economic  significance  of  the 
machine  industry  is  only  beginning  to  receive  from 
economists  and  educators  the  consideration  it  deserves. 
Laws  on  manufacturing  conditions  and  on  child  labor 
indicate  a  growing  appreciation  on  the  part  of  the 


10  THE  REALITIES  OF  MODERN  SCIENCE 

public  and  its  statesmen  of  a  few  of  the  problems 
involved. 

With  the  control  over  fire,  and  the  consequent  ability 
to  cook  food  and  to  maintain  a  comfortable  living 
temperature,  man's  ascent  toward  civilization  really 
began.  During  this  time  he  devised  and  employed 
many  simple  machines,  some  of  which  we  shall  describe 
in  a  later  chapter.  From  living  in  crude  shelters  of 
boughs  and  leaves  or  in  caves  he  came  to  live  in  tents 
and  houses.  From  being  dependent  on  wild  fruits 
and  meats  he  advanced  to  agriculture.  Animals  of 
burden  were  used  as  machines  with  internal  sources 
of  energy.  Boats  were  made  and  fitted  with  oars, 
which  are  of  course  simple  machines,  and  also  with  sails 
by  means  of  which  the  energy  of  the  wind  could  be 
utilized.  Larger  groups  of  men  could  now  exist  in 
the  same  region,  and  cities  and  villages  were  formed. 
With  the  lessening  of  the  struggle  for  existence  man 
obtained  more  time  for  thought  and  mental  develop- 
ment. The  large  empires  of  the  ancient  world  and 
their  civilizations  were  founded  on  these  elementary 
advances  in  the  knowledge  of  the  physical  world. 
Language  and  the  arts  had  developed  so  that  with  two 
of  these  empires,  Babylon  and  Egypt,  history  as  we 
know  it  really  began.  With  these  there  also  began 
that  arrangement  and  correlation  of  knowledge  which 
we  call  science. 

Both  the  Babylonians  and  the  Egyptians  were 
agricultural  peoples,  one  living  in  the  fertile  valley  of 
the  Euphrates,  the  other  on  the  banks  of  the  Nile. 
What  scientific  advances  they  made  were  mostly  along 
lines  of  immediate  practical  value.  Thus,  the  Egyp- 


THE  BEGINNINGS  OF  KNOWLEDGE  11 

tians  are  believed  to  have  learned  mensuration  and 
started  the  science  of  "earth  measurements, "  or  geom- 
etry, because  of  the  necessity  of  marking  anew  each 
year  the  boundaries  of  fields  flooded  by  the  Nile.  They 
did  not  carry  their  development,  however,  beyond  the 
simple  ideas  necessary  to  lay  off  plots  of  ground  and 
to  compute  areas  and  volumes.  It  was  left  for  the 
Greeks  to  develop  later  the  abstract  ideas  and  general 
methods  of  reasoning  which  we  know  to-day  as  geom- 
etry. 

Both  these  peoples  were  highly  religious.  With 
then-  lack  of  scientific  knowledge  it  was  but  natural 
that  they  should  be  superstitious  about  the  stars,  and 
that  their  knowledge  of  astronomy  should  develop  hi 
connection  with  their  religion.  To  the  Babylonians 
we  apparently  owe  the  grouping  of  days  into  longer 
periods  called  weeks.  Why  seven  days  were  chosen 
for  the  latter  period  is  evident  from  our  names  for  these 
days,  which  are  traced  in  derivation  to  the  sun,  the 
moon,  and  the  five  planets,  Jupiter,  Mercury,  Mars, 
Venus  and  Saturn.1  The  lunar  month  was  of  course 
early  noticed  and  measured  by  these  astronomers. 
They  had  also  progressed  far  enough  to  measure  the 
year  as  365i  days  and  to  determine  other  astronomical 
facts. 

These  facts  were  developed  in  connection  with  their 
religion  and  their  practical  needs.  In  Babylonia, 
where  stone  was  not  easily  obtainable  and  brick  was 
used  for  building,  one  of  the  months  was  named  for 
the  brickbuilders.  In  this  month,  apparently,  the 

1  Cf.  Libby,  "An  Introduction  to  the  History  of  Science," 
Houghton  Mifflin  Company,  1917. 


12  THE  REALITIES  OF  MODERN  SCIENCE 

clay  and  the  atmosphere  best  suited  this  work.  In 
Egypt,  on  the  other  hand,  great  interest  was  taken  in 
the  various  constellations  in  which  the  sun  appeared 
at  dawn.  The  heliacal  rising  of  Sirius  marked  the 
annual  inundation  of  the  lands  adjacent  to  the  Nile 
and  began  the  calendar  year. 

It  is  interesting  to  note,  also,  that  the  division  of  hours 
and  minutes,  each  into  sixty  parts,  may  be  traced  to  the 
Babylonians.  Although  they  had  a  decimal  system 
of  numbers,  they  also  had  a  sexagesimal  system.  In 
their  system  of  weights,  starting  with  a  "shekel" 
equivalent  to  about  half  an  ounce,  they  had  a  unit, 
the  "mina,"  sixty  times  as  large,  and  also  a  unit,  the 
"talent/'  which  was  equal  to  sixty  minas. 

In  general,  both  these  early  races  were  singularly 
lacking  in  curiosity  as  to  the  world  in  which  they  lived, 
as  to  its  natural  laws,  its  origin,  and  its  composition. 
They  did,  however,  invent  new  contrivances  and  thus 
showed  a  practical  familiarity  with  some  of  the  phe- 
nomena of  nature.  As  early  as  4000  B.C.  the  Egyptians 
had  learned  to  melt  iron  oy  using  a  bellows  to  make 
the  fire  burn  hot  enough. 

In  some  respects  their  temperament  was  like  our  own. 
They  were  interested  in  practical  applications  and  were 
unconcerned  or  even  impatient  of  attempts  to  classify 
and  coordinate  their  knowledge  and  to  speculate  upon 
the  fundamental  causes  of  the  phenomena  which  they 
observed.  They  did  not  realize  that  practical  devices  fol- 
low and  do  not  in  general  precede  advances  in  scientific 
knowledge.  In  the  case  of  the  Egyptians  this  lack  of 
speculative  interest  was  combined  with  a  feeling  that  in 
their  religion  they  had  reached  a  finality.  As  a  result 


THE  BEGINNINGS  OF  KNOWLEDGE  13 

there  was  a  long  period  in  which  they  made  small 
progress  and  during  which  they  were  passed  by  other 
races. 

The  rise  of  one  of  these,  the  Greek,  is  marked  in 
its  early  stages  by  contributions  from  the  Semitic  race, 
of  which  the  Babylonians  and  Egyptians  were  both 
members.  The  Island  of  Crete  first  received  the 
culture  of  the  Semites  from  the  Egyptian,  Assyrian, 
and  Phoenician  sailors  and  merchants.  But  the  Greeks 
also  were  intrepid  sailors  and  shrewd  merchants,  and 
they  were  shortly  to  be  found  in  the  older  centers  of 
learning.  With  one  of  these  Greeks,  an  Ionian  of 
Phoenician  descent,  Thales  of  the  Island  of  Miletus, 
science  as  well  as  Greek  philosophy  is  considered  to 
have  begun.  This  was  about  585  B.C. 


CHAPTER  II 

THE  MACHINES  OF  THE  ANCIENT  WORLD 

IN  the  homes  of  the  earlier  civilizations  are  buildings 
and  monuments  which  have  withstood  the  ravages  of 
time  and  the  waves  of  hostile  invasion.  Often  these 
records  of  man's  earlier  activities  are  only  restored  to 
sight  by  excavation,  for  they  have  been  covered  by  the 
dust  of  succeeding  ages.  The  temples  and  palaces, 
the  city  walls,  and  the  monuments  of  Egypt  and  Asia 
were  of  stone  or  brick,  but  the  houses  of  the  common 
people  were  usually  of  mud.  When  a  conqueror  razed 
a  city,  holes  were  broken  in  the  walls  to  render  it  less 
easily  defended ;  the  monuments  of  the  kings  and  the 
altars  of  the  local  gods  were  toppled  over;  and  the 
houses  of  the  common  people  were  reduced  to  dust. 
The  large  buildings,  of  course,  took  years  to  restore, 
and  the  weakened  and  terrorized  inhabitants  may  have 
hesitated  to  rebuild  what  had  so  recently  been  de- 
stroyed by  their  enemy.  Thus  in  time  the  traces  of  a 
city  were  lost  beneath  the  mud  and  rubbish  of  succeed- 
ing generations. 

Such  relics  as  we  have,  suggest  to  the  engineer  of 
to-day  the  question  :  How  did  the  ancients,  with  their 
limited  knowledge  of  physical  science,  succeed  in 
erecting  such  enormous  and  high  structures?  The 
answer  is  to  be  seen  partly  in  their  knowledge  of  certain 

14 


THE  MACHINES  OF   THE  ANCIENT  WORLD        15 


simple  machines  and  partly  in  their  organization  of 
society.  They  were  familiar,  we  know,  with  some  or 
all  of  six  mechanical  contrivances,  known  as  the  six 
simple  machines.  These  are  the  lever,  the  inclined 
plane,  the  wedge,  the  pulley,  the  screw,  and  the  wheel 
and  axle,  conventionally  illustrated  hi  Fig.  3.  Let 
us  see  what  the  advantages  of  these  devices  would  be. 

The  lever  consists  of  a  bar.     If  the  distance  from  the 
fulcrum,  Cj  to  the  point  a,  where  the  force  is  applied, 


FIG.  3.. 

is  larger  than  that  to  the  point  b,  then,  as  we  know,  the 
lever  gives  a  mechanical  advantage.  The  force  ex- 
erted at  6  by  the  lever  is  as  many  times  greater  than 
the  force  applied  at  a  as  the  distance  ac  is  greater  than 
be.  It  was  not  until  the  time  of  Archimedes  (died 
212  B.C.)  that  men  knew  this  rule.  During  preceding 
centuries,  however,  they  must  have  had  a  very  good 
working  knowledge  of  the  lever. 
When  a  man  wanted  to  raise  the  end  of  a  block  of 


16  THE  REALITIES  OF  MODERN  SCIENCE 

stone  he  probably  selected  a  bar  which  looked  about 
long  enough  and  tried.  If  it  didn't  work  he  pushed 
the  fulcrum  a  little  closer  to  the  block  and  tried  again. 
If  the  force  he  was  capable  of  applying  was  still  too 
small,  he  tried  a  longer  lever.  He  thus  made  the  lever 
arm  of  his  force  sufficiently  greater  than  that  of  the 
weight  he  was  trying  to  lift.  Of  course,  if  he  chose  too 
long  a  bar  it  may  have  bent  in  such  a  way  as  really  to 
make  his  lever  arm  shorter.  In  some  such  manner  the 
ancients  probably  obtained  their  knowledge  of  the  lever 
and  incidentally  some  ideas  as  to  the  bending  and 
elasticity  of  beams. 

We  know  to-day  another  property  of  the  lever  of  which 
they  made  use,  although  it  was  not  formally  and  com- 
pletely described  in  words  for  centuries.  We  recognize 
that  the  motion  of  a  lever  is  a  rotation  about  its  fulcrum, 
the  path  which  any  point  travels  being  a  part  of  a  circle 
about  the  fulcrum.  Obviously,  in  any  motion,  the 
more  distant  points  must  move  through  the  greater 
distances.  But  the  time  it  takes  each  point  to  move 
through  the  arc  of  its  circle  is  the  same.  Hence  the 
point  that  moves  the  farther  must  also  move  the  faster. 
This  is  the  property  which  we  use  whenever  we  want  to 
strike  faster  than  we  can  with  the  levers  afforded  us  in 
the  bones  of  our  bodies,  as  in  using  an  ax,  a  sword,  or 
a  golf  club.  The  system  of  levers  involved  in  an 
ordinary  typewriter  is  an  interesting  illustration  of 
a  slow  motion  of  the  point  where  the  force  is  applied 
producing  a  faster  motion  of  another  point  of  the 
system. 

In  the  case  of  the  lever,  as  of  all  machines,  work  is 
done  by  the  machine  only  because  work  is  done  on  it. 


THE  MACHINES  OF  THE  ANCIENT  WORLD   17 

Now,  we  know  that  when  we  push  or  pull,  the  work 
we  do  depends  not  only  upon  how  hard  but  upon  how 
far.  When  one  pushes  on  the  long  arm  of  a  lever  and 
lifts  a  larger  weight  at  the  end  of  the  short  arm,  he 
must  push  a  correspondingly  greater  distance  than  the 
weight  rises. 

The  inclined  plane  is  strictly  speaking  not  a  machine. 
Its  serviceability  was  probably  evident  to  man  fairly 
early  in  his  development.  He  would  have  noticed 
that  he  could  drag  a  burden  up  the  smooth  inclined 
face  of  a  rock  more  easily  than  lift  it  through  the  same 
vertical  distance.  lii  his  building  operations  he  learned 
something  of  what  builders  to-day  call  the  "  angle  of 
repose"  of  a  loose  material.  When  we  try  to  make 
a  pile  of  sand  or  gravel  we  notice  that  if  the  sides 
become  too  steep,  what  we  put  on  slips  down.  If 
we  do  not  exceed  a  certain  steepness,  that  is,  a  certain 
angle  of  repose,  then  the  friction  which  the  surface 
offers  prevents  what  we  add  to  the  pile  from  sliding 
down.  The  steeper  the  inclined  plane,  the  more  easily 
do  things  slide  down  and  the  greater  the  difficulty 
of  dragging  them  up.  For  this  reason  the  slope  of 
the  incline  should  be  gradual.  It  is  also  desirable 
to  reduce  as  much  as  possible  the  friction  between  the 
body  and  the  surface  of  the  incline.  Of  course,  on 
roads  this  is  usually  accomplished  by  making  the 
friction  between  the  two  surfaces  rolling  rather  than 
sliding  friction.  With  these  ideas  the  ancients  were 
familiar. 

It  is  our  common  knowledge  as  to  friction  that  it  is 
greater  the  harder  the  two  sliding  surfaces  are  pressed 
together  and  the  more  irregularities  there  are  upon 


18  THE  REALITIES  OF  MODERN  SCIENCE 

them.  Automobile  tires  have  specially  shaped  surfaces 
or  chains.  Because  of  the  fact  that  the  amount  of 
friction  which  will  oppose  the  action  of  a  sliding  body 
depends  upon  the  character  of  the  two  surfaces,  it  is 
easier  when  we  wish  to  develop  the  law  for  an  inclined 
plane  to  consider  first  a  case  where  there  is  no  friction. 
But  such  a  case  is  impossible ;  even  between  the  most 
highly  polished  and  oiled  surfaces  there  would  be  some 
friction.  Nevertheless  it  is  a  method  of  science  to 
consider  first  an  " ideal  problem/'  "the  limiting  case" 
of  the  general  problem.  Thus  in  the  present  instance, 
of  all  the  possible  cases  of  bodies  on  inclined  planes, 
we  decide  to  discuss  first  that  case  which  lies  just  at 
the  limit  of  the  physically  possible,  where  the  friction 
is  so  small  that  we  may  entirely  neglect  it. 

Consider  the  work  of  moving  a  body  from  one  end 
a  to  the  other  end  c  of  the  inclined  plane  shown  in 
Fig.  3.  Experience  tells  us  that  as  we  push  the  body 
upward  the  plane  supports  part  of  its  weight.  If  there 
is  no  frictional  opposition  to  sliding,  the  work  we  do 
is  that  of  lifting.  In  sliding  it  along  the  side  ac  we  do 
not  have  to  push  as  hard,  but  we  must  push  farther,  so 
that  we  do  just  as  much  work  as  if  we  lifted  the  body 
vertically  through  the  distance  be.  In  both  the  in- 
clined plane  and  the  lever  the  principle  is  the  same, 
namely,  a  small  force  exerted  for  a  long  distance  does 
the  same  work  as  a  larger  force  for  a  shorter  distance. 

The  wedge  is  a  sort  of  double  inclined  plane.  Of 
all  the  elementary  machines  it  was  probably  the  earliest, 
for  it  is  really  a  cutting  tool  like  a  chisel.  Some  primi- 
tive man  in  trying  to  scrape  a  hole  through  a  block  of 
wood  by  using  a  sharp  stone  may  have  got  it  stuck  in 


THE  MACHINES  OF   THE  ANCIENT   WORLD       19 

the  wood  and  while  pounding  it  out  split  the  block. 
This  use  of  a  wedge  for  prying  objects  apart  is  so  old  as 
to  be  almost  instinctive.  Where  a  wedge  is  driven  in 
by  pounding,  friction  is  very  desirable,  for  if  there  were 
no  friction  it  would  drive  hi  somewhat  more  easily  but 
it  would  slip  back  and  out  following  each  blow. 

If  one  cuts  a  long  right  triangle  of  stiff  paper  and 
then  winds  it  about  a  pencil,  he  will  see  how  a  screw 
is  but  a  special  form  of  an  inclined  plane.  Resting 
a  finger  nail  lightly  on  the  " thread"  of  the  screw  he 
may  observe  how  turning  the  screw  slides  the  finger 
up  the  spiraled  incline.  In  practical  applications  of 
this  principle  the  screw  turns  in  a  nut.  If  the  nut  is 
prevented  from  turning  while  the  screw  is  turned  the 
rotation  of  the  screw  will  result  hi  its  translation  rel- 
ative to  the  nut.  The  screw  offers  a  large  mechanical 
advantage,  to  which  friction,  however,  imposes  a 
practical  limit. 

The  pulley  followed  in  the  development  of  man's 
mechanical  ability  his  invention  of  rope.  For  rope 
he  may  first  have  used  long  vines,  and  later,  using  two 
or  more  together  to  obtain  greater  strength,  learned 
to  twist  rope.  Considerable  invention,  however,  was 
required  to  produce  a  pulley,  although  the  idea  of 
changing  the  direction  hi  which  a  force  is  exerted  by 
passing  the  rope  over  the  limb  of  a  tree  was  perhaps 
obtained  by  the  easy  accident  of  pulling  at  some  vines 
which  grew  over  a  tree.  Between  such  an  observation 
as  this  and  the  idea  of  reducing  the  friction  by  allowing 
the  bar  over  which  the  rope  passed  to  roll,  there  is  a 
large  step.  The  idea  that  round  objects  roll  more 
easily  than  they  slide  may  have  come  from  noticing 


20  THE   REALITIES  OF   MODERN   SCIENCE 

pebbles  and  bowlders  rolling  down  a  hillside,  or  from 
the  turning  of  bits  of  branches  under  foot.  From  the 
use  of  rollers  to  wheels  was  possibly  the  line  evolution 
took.  From  wheels  to  pulleys  is  another  possible 
step.  A  wheel,  rolling  along  the  ground,  comes  into 
contact  with  successive  points,  and  the  rotating 
axle  moves  forward.  In  the  pulley  successive  points 
of  the  rope  pass  from  one  side  to  the  other  of  a  rotating 
axle.  In  both  cases  the  rotation  of  the  wheel  about 
its  axis  is  accompanied  by  a  translation  of  the  con- 
tacting surface  relative  to  the  axle. 

The  pulley  was  of  special  value  in  early  building 
operations  because  there  were  no  sources  of  energy, 
like  steam  engines,  which  could  exert  large  forces. 
Large  forces  were  derived  from  a  large  number  of  men  ; 
and  a  pulley  allowed  their  combined  force  to  be  exerted 
in  the  desired  direction.  It  is  important  to  notice 
that  with  a  single  pulley  the  only  effect  is  that  of 
changing  the  direction  of  the  force.  With  a  combina- 
tion of  pulleys  we  may  obtain  mechanical  advantage 
which  was  apparently  unknown  to  the  ancients.1  In 
such  a  case  we  arrange  the  system  so  that  the  weight 
to  be  lifted  moves  a  smaller  distance  than  the  end  of 
the  rope  which  is  being  pulled.  In  this  way  we  make 
use  of  the  principle  which  applies  to  the  lever  and  the 
inclined  plane.  This  "work  principle"  states  that  in 
any  machine  the  work  of  the  acting  force  is  equal  to 
the  work  done  against  the  resisting  force.  The  work 
in  each  instance  is  measured  by  the  product  of  the 
force  and  the  distance. 

1  The  story  of  page  43  indicates  that  Archimedes  was  familiar 
with  it,  but  this  was  much  later  than  the  building  of  the  pyramids. 


THE  MACHINES  OF   THE  ANCIENT   WORLD       21 

The  wheel  and  axle  is  something  like  a  pulley.  It 
permits  a  mechanical  advantage,  since  the  distance 
through  which  a  point  on  the  wheel  moves  may  be 
large  as  compared  to  the  corresponding  distance 
through  which  the  weight  moves. 

Of  course,  in  the  case  of  all  these  machines,  some 
of  the  work  of  the  operator  is  done  against  friction, 
and  appears  as  heat  at  the  point  where  the  friction 
occurs.  The  work  which  the  operator  gets  out  of  the 
machine  is  then  less  than  he  puts  in.  The  ratio  of 
the  useful  work  to  the  input  is  the  efficiency.  In  the 
ideal  case  it  is  100%,  but  no  actual  machines  have 
efficiencies  near  this  limiting  value. 

With  these  simple  contrivances  the  buildings  of  the 
ancient  world  were  constructed.  We  do  not  know 
the  exact  methods  followed  by  their  erectors,  but  it 
is  possible  that  they  used  derricks  to  set  the  stones. 
In  addition,  as  the  building  progressed,  they  used 
scaffolding.  For  the  Babylonian  structures,  it  appears, 
these  scaffoldings  were  of  brick.  They  would  need 
to  be  quite  substantial  to  support  the  large  weights 
of  men  and  materials  necessary  for  such  enormous 
buildings  as  we  have  seen  in  the  figures.  To-day  frame 
scaffoldings  may  be  made  of  comparatively  light  beams. 
Before  man  learned  the  underlying  physical  laws, 
strength  and  stability  could  be  obtained  only  by 
massive  construction,  of  which  the  Pyramids  form  per- 
haps the  most  striking  illustration. 

The  energy  to  operate  such  machines  and  through 
them  to  rear  such  structures  was  not  derived  from 
natural  sources  through  inanimate  means.  Of  the  use 
of  winds  the  ancients  had  some  knowledge,  but  except 


22  THE   REALITIES   OF   MODERN   SCIENCE 

for  sailing  boats  or  for  operations  where  continuity  is 
not  essential,  wind  is  not  a  satisfactory  source  of  energy. 
Horses,  it  is  true,  were  available,  but  it  is  doubt- 
ful if  they  were  much  used  for  such  purposes  as  these. 
They  were  not  the  property  of  the  poorer  classes,  and 
were  used  for  hunting  and  for  warfare.  For  these 
buildings  the  energy  was  derived  from  men.  Beside 
one  of  the  large  pyramids  of  Gizeh  stand  the  quarters 
of  the  workmen,  long  chambers,  capable  of  housing 
4000  men.  When  we  are  told  that  the  roof  over  the 
chamber  in  one  of  the  pyramids  was  formed  by  three 
layers  of  cut  stones,  each  piece  weighing  about  30 
tons,  we  realize  more  fully  the  large  amount  of  human 
energy  which  must  have  been  expended  in  such  a  con- 
struction. The  laborers  who  supplied  this  energy 
were  slaves,  usually  members  of  subject  races,  who 
performed  such  tasks  much  as  the  Hebrews  once  made 
brick  for  the  Egyptians. 

It  would  be  of  interest  to  follow  the  development  of 
slavery  to  its  peak  in  the  Roman  Empire,  its  transi- 
tion into  feudalism,  and  the  organization  of  society  in 
medieval  Europe.  Through  all  the  centuries  from  the 
earliest  Egyptian  records  to  the  present  time,  under 
one  form  or  another  of  slavery,  men  have  supplied  the 
energy  for  the  monuments  of  dynasties.  When  man's 
development  along  scientific  lines  was  yet  too  small 
to  allow  him  to  utilize  the  enormous  stores  of  energy 
in  the  world  about  him,  the  main  source  of  energy  was 
in  food,  for  the  conversion  of  which  into  mechanical 
energy  the  human  body  is  an  efficient  engine.  But 
to-day  man's  growing  scientific  abilities  are  making 
possible  grandeurs  of  buildings  and  comfort  of  builders 


THE  MACHINES  OF  THE  ANCIENT  WORLD   23 

far  in  excess  of  the  dreams  of  the  ancients,  and  his 
growing  spirit  of  democracy  is  insisting  that  human 
energy  shall  not  be  wasted  and  that  the  energy  of  the 
material  world  shall  be  efficiently  utilized  for  the  com- 
fort of  all. 


CHAPTER  III 

WEIGHTS  AND   MEASURES 

IT  has  been  well  said  by  one  of  the  leading  physicists 
of  the  19th  century  that  "when  you  can  measure  what 
you  are  speaking  about  and  express  it  in  numbers  you 
know  something  about  it,  but  when  you  can  not,  your 
knowledge  is  of  a  meager  and  unsatisfactory  kind ; 
it  may  be  the  beginning  of  knowledge,  but  you  have 
scarcely  in  your  thoughts  advanced  to  the  stage  of 
science. "  In  later  chapters  we  shall  see  how  science 
began,  and  learn  something  of  the  uses  of  mathe- 
matics. 

To  tell  "how  much,"  we  must  have  some  means  for 
measuring,  and  a  unit.  For  example,  to  measure  a 
length  we  take  some  length  as  a  unit,  that  is,  call  some 
definite  length  "one."  We  may  then  count  how  many 
times  this  unit  goes  into  the  unknown  length.  The  unit 
we  use  for  this  purpose  may  be  a  length  of  any  sub- 
stance, as  of  string  between  two  knots,  for  the  material 
of  the  standard  is  unimportant  provided  it  is  not  such 
as  to  change  its  length  while  we  are  measuring  with 
it.  If  a  boy  wished  a  measure  of  how  fast  he  was 
growing  he  might  record  his  height  each  year  on  some 
upright.  He  would  not,  however,  choose  a  poplar 
sapling. 

This  illustration  makes  the  use  of  a  variable  standard 

24 


WEIGHTS  AND  MEASURES  25 

seem  very  foolish,  l  but  such  a  possibility  must  always 
be  guarded  against,  especially  when  precise  measure- 
ment is  desired.  For  example,  a  steel  tape  line  varies 
in  its  length,  increasing  on  hot  days.  In  surveying 
over  large  areas,  as  in  the  case  of  the  U.  S.  Coast  and 
Geodetic  Surveys,  extreme  precautions  are  taken  in 
measuring  a  base  line  to  compensate  for  such  linear 
expansion.  If  a  tape  is  used  care  is  also  taken  that  it 
shall  always  be  under  the  same  tension. 

For  the  measurement  of  length  it  was  natural  that 
the  earliest  units  should  be  connected  with  the  human 
body.  Thus  lengths  were  measured  in  spans,  in  digits, 
in  feet,  or  in  the  length  of  a  step  or  pace.  Longer 
distances  were  measured  in  terms  of  a  day's  march  or 
journey.  These  units  all  differ  from  man  to  man, 
so  that  if  two  persons  measure  the  same  distance  they 
will  express  it  differently,  because  they  will  count 
different  numbers  of  units. 

Thus  it  came  about  when  men  wished  to  have  a 
common  understanding  that  they  had  to  agree  as  to 
whose  foot  or  pace  should  be  used  as  the  standard. 
Many  times  these  lengths  were  referred  to  the  body 
of  a  king  or  of  a  priest.  In  order  that  the  chosen  units 
should  be  commonly  available  for  comparison,  they 
were  sometimes  marked  on  the  wall  of  the  city  or  of 
a  temple.  This  method  of  recording  standards  per- 
sisted until  comparatively  recently,  for  there  are 
European  cathedrals  on  the  outer  walls  of  which 
standards  are  marked. 

1  The  use  in  measurements  of  a  standard  which  changes,  without 
making  proper  allowance  for  such  changes,  we  recognize  as  a  form 
of  mistake  which  is  frequently  made  in  the  popular  discussions  of 
social  and  economic  relations. 


26  THE   REALITIES  OF   MODERN  SCIENCE 

The  standards  would  sometimes  be  lost  or  destroyed, 
and  then  a  new  unit  would  have  to  be  chosen.  Per- 
haps the  new  unit  was  made  practically  of  the  same 
length  as  the  old  by  reference  to  the  walls  of  some 
building  for  which  people  remembered  the  lengths 
as  expressed  in  the  former  unit.  At  any  rate  it  is  by 
this  method  that  archeologists  to-day  obtain  their 
ideas  as  to  the  length  of  the  units  used  in  ancient  times. 

Other  units  may  be  formed  by  taking  multiples  and 
submultiples  of  the  adopted  unit.  This  is  also  the  custom 
to-day  in  the  case  of  the  so-called  metric  units,  of  which 
the  meter  is  the  unit  of  length.  The  multiple  unit  which 
occurs  most  frequently  is  formed  by  the  prefix  "kilo, " 
meaning  "one  thousand."  The  submultiples  are 
formed  by  using  Latin  prefixes  instead  of  Greek,  thus 
"centi"  and  "milli"  for  "one  hundredth"  and  "one 
thousandth." 

Different  localities  had  different  units,  although  there 
was  a  tendency  toward  the  reduction  of  the  number  of 
units,  due  to  the  formation  of  large  empires  and  to  the 
usages  of  traders.  Such  a  standardization  ultimately 
resulted  in  greater  simplicity  and  facilitated  trade, 
for  we  all  recognize  the  inconvenience  attaching  to 
the  use  of  different  systems,  e.g.  metric  and  English. 
The  tendency  to  standardize  was  most  marked  in  the 
case  of  units  of  weight.  If  money  is  to  perform  its 
proper  service  in  the  exchange  of  goods,  then  each  piece 
of  a  given  denomination  must  be  as  nearly  as  possible 
the  same  as  any  other.  It  was  somewhat,  probably, 
to  insure  the  full  payment  of  taxes  and  for  reasons 
of  pride,  but  largely  to  stabilize  trade,  that  monarchs 
were  especially  concerned  with  weights  and  measures. 


WEIGHTS  AND  MEASURES  27 

Of  course  a  dishonest  monarch  might  debase  his  own 
currency  by  retaining  the  same  form  and  markings 
for  a  com  but  decreasing  the  amount  of  precious  metal 
in  it,  and  thus  be  enabled  for  a  time  to  purchase  from 
his  own  people  or  from  unsuspecting  traders  a  greater 
amount  of  goods  for  the  same  metal.  Foreign  traders, 
however,  are  usually  alertly  suspicious,  so  that  only 
the  full  weight  coin  would  circulate  in  foreign  trade,  as 
the  reader  realizes  from  his  familiarity  with  Gresham's 
law. 

For  standards  of  weight,  pieces  of  metal  or  stone 
were  used.  Sometimes  these  were  in  the  form  of 
animals,  as  the  bronze  lion  of  the  Assyrians.  In 
England  one  of  the  standards  of  weight  was  known 
from  its  material  as  "the  stone."  The  use  of  this 
unit  has  persisted  for  years  and  is  still  evident  in 
expressions  for  the  weight  of  a  man.  To-day  in  England, 
hi  the  British  Commonwealth,  except  for  India,  and 
also  in  the  United  States  the  unit  of  weight  is  the 
pound.  In  the  rest  of  the  countries  of  the  world  the 
unit  is  the  kilogram  of  the  metric  system.  To  the 
scientist  this  system  is  the  more  important. 

While  in  the  case  of  a  measurement  of  length  it  is 
only  necessary  to  set  the  standard  beside  the  body  of 
unknown  length  to  make  a  direct  comparison  of  the 
two,  for  measurement  of  weight  special  apparatus  is 
required.  For  weighing  it  is  necessary  to  have  a  set 
of  weights,  comprising  multiples  x  and  submultiples 

1  A  set  of  weights  may  be  obtained  as  follows  when  one  has  ac- 
cess to  a  balance  and  the  standard  weight.  First  it  is  usual  to  make 
a  copy  of  the  original  standard,  or  "prototype,"  for  use  in  place  of 
it.  This  is  made  a  little  too  heavy  to  start  with  and  reduced  until 
it  is  as  nearly  as  possible  equal  to  the  standard.  The  copy  is  then 


28  THE   REALITIES  OF   MODERN   SCIENCE 

of  the  unit  weight.  With  these  and  a  balance  the 
unknown  weight  of  a  body  may  be  determined  in  an 
obvious  manner.  The  balance  is  merely  a  lever 
supported  at  the  center.  At  the  ends  of  the  equal 
lever  arms  thus  formed,  pans  are  hung,  in  which  the 
weights  to  be  compared  are  placed.  With  this  balance 
the  ancients  were  familiar  and  Pythagoras  in  the  sixth 
century  B.C.  notes  that  dishonest  tradesmen  used  to 
shift  the  fulcrum  slightly  away  from  the  pan  containing 
what  they  were  selling  so  that  a  false  comparison 
resulted  in  their  favor.  Of  course  the  check  against  such 
a  fraud  is  to  interchange  the  contents  of  the  two  pans, 
for  then  any  inequality  will  be  at  once  evident.  In 
general,  if  the  arms  are  unequal  the  true  weight  of  the 
body  may  be  obtained  by  observing  the  weights  re- 
quired to  balance  it,  first  in  one  pan  and  then  in  the 
other.  This  method  is  called  " double  weighing." 
(From  these  two  weighings  the  true  weight  is  obtainable 
as  the  geometric  mean.) 

Another  method  of  obtaining  the  true  weight  of  a 
body,  even  though  the  lever  arms  are  unequal,  is  known 
as  the  "  substitution  method."  The  unknown  body 
is  placed  in  one  pan  and  balanced  by  the  addition  to 
the  other  pan  of  lead  shot,  sand,  or  any  convenient 

used  for  a  standard  and  the  prototype  is  preserved  for  occasional 
comparisons. 

To  obtain  a  weight  of  two  units  it  is  only  necessary  to  make  an- 
other weight  like  the  copy  and  then  to  make  a  new  weight  which 
will  just  balance  the  other  two.  Two  half  units  may  be  obtained 
in  the  following  manner.  Weights  of  approximately  half  a  unit 
are  made.  These  are  balanced  against  each  other  and  made  equal. 
They  are  then  kept  equal  and  worked  down  until  together  they 
just  balance  the  unit.  In  somewhat  similar  manner  other  sub- 
multiples  may  be  obtained. 


WEIGHTS  AND  MEASURES  29 

substance.  After  a  balance  is  obtained  known  weights 
are  substituted  for  the  unknown  body  until  a  balance 
is  again  obtained.  It  is  evident  that  these  known 
weights  produce  the  same  effect  under  the  same  con- 
ditions as  does  the  body  of  unknown  weight  and  hence 
that  they  are  equivalent.  The  method  requires  that 
everything  else  which  can  hi  any  way  affect  the  actions 
to  be  compared  must  remain  unchanged,  or  in  the 
usual  words,  all  other  factors  must  be  constant. 

We  are  now  ready  to  give  a  name  to  the  method  of 
direct  comparison,  illustrated  by  the  use  of  the  balance. 
If  the  lever  arms  are  equal,  equal  weights  produce 
equal  and  opposite  effects  which  neutralize  each 
other,  so  that  there  is  no  deflection  of  the  beam.  This 
is  therefore  called  the  " opposition  method,"  the  "zero 
deflection  method,"  or  the  "null  method"  from  the 
Latin  "nullus."  It  is  the  method  of  no  resultant 
effect. 

These  two  methods,  the  opposition  and  the  sub- 
stitution, are  the  only  ones  by  which  we  can  compare 
two  things,  whether  weights,  electric  currents,  light 
intensities,  or  any  of  the  other  magnitudes  which  the 
scientist  may  have  occasion  to  measure.  The  method 
of  double  weighing  is  evidently  merely  the  opposition 
method  used  twice  with  the  weights  reversed,  so  that 
the  effect  of  inequalities  in  the  lever  arms  of  the  balance 
may  be  eliminated. 

The  advantages  of  the  two  methods  are  now  evident. 
With  the  opposition  method  a  direct  comparison  is 
obtained  by  one  operation,  while  with  the  substitution 
method  two  operations  are  required.  On  the  other 
hand,  if  the  possible  inequalities  in  the  two  parts  of  the 


30  THE   REALITIES  OF   MODERN  SCIENCE 

apparatus,  through  which  opposing  effects  are  balanced, 
are  to  be  eliminated  by  a  reversal,  as  in  double  weighing, 
the  number  of  operations  required  for  equally  accurate 
results  in  the  two  methods  becomes  the  same.  For 
many  measurements,  as  for  example  in  astronomy, 
it  is  impossible  to  apply  the  substitution  method  and 
results  of  marvelous  accuracy  are  obtained  by  the 
opposition  method. 

Both  the  substitution  method  and  the  opposition 
method  with  reversals,  are  inconvenient  because  they 
take  so  much  time,  as  is  evident  from  the  illustration 
of  weighing.  To-day  the  opposition  method  of  weigh- 
ing is  used  only  where  especially  precise  results  are 
required,  as  for  example,  in  chemical  analysis,  phar- 
maceutical work,  and  in  the  weighing  of  precious  stones 
and  metals.  For  ordinary  purposes  the  use  of  spring 
scales  makes  possible  a  more  convenient  form  of  sub- 
stitution method. 

A  spring  scale  depends  for  its  operation  upon  elas- 
ticity,1 which  is  "the  property  by  virtue  of  which  a 
body  requires  the  continued  application  of  a  deform- 
ing stress  to  prevent  the  recovery,  entire  or  partial, 

1  This  definition,  as  the  reader  recognizes,  is  a  good  example  of 
the  expression  of  an  abstract  idea.  Thus  the  use  of  the  word 
"body"  meaning  a  "definite  amount  of  matter"  does  not  limit  the 
definition  to  any  particular  substance  or  form.  Again  in  the  word 
"deforming"  we  get  away  from  the  concrete  by  using  a  word  ex- 
pressing an  abstract  idea,  for  it  covers  stretching,  bending,  twisting, 
compressing,  distending,  in  fact,  any  change  in  form.  To  stretch 
we  must  pull  in  opposite  ways,  to  compress  we  push  from  opposite 
sides,  to  distend  we  push  outward  in  every  direction.  A  single 
force  will  not  produce  a  deformation.  To  convey  the  idea  of  com- 
binations of  oppositely  directed  equal  forces  the  physicist  uses  the 
word  "stress." 


WEIGHTS  AND  MEASURES  31 

from  deformation."  The  physicist  also  uses  the  word 
" strain,"  to  represent  the  measure  of  the  deformation 
which  a  stress  produces.  This  happens  to  make  it 
easier  to  remember  the  fundamental  law  of  elasticity, 
namely :  The  ratio  of  stress  to  strain  is  constant. 

In  our  definition  for  elasticity  we  see  that  the  re- 
covery when  the  deforming  stress  is  removed  may  be 
" entire  or  partial."  If  it  is  entire  the  body  is  perfectly 
elastic.  One  of  the  best  illustrations  of  perfect  elas- 
ticity is  the  hairspring  of  a  watch,  which  may  coil  and 
uncoil  a  million  times  without  any  permanent  deforma- 
tion. If,  however,  a  body  is  deformed  too  far  it  does 
not  return  to  its  original  form,  although  it  may  still 
show  some  ability  to  recover  from  the  deforming  stress. 
This  is  imperfect  elasticity.  When  the  stress  becomes 
so  large  that  the  body  does  not  recover  entirely,  the 
" elastic  limit"  is  said  to  have  been  reached,  and  for 
greater  stresses  the  alliterative  law  given  above  no 
longer  holds.  For  some  still  larger  stress  the  body 
breaks.  This  breaking  stress  measures  the  "  ultimate 
strength"  of  the  body. 

With  the  quantitative  applications  of  these  prin- 
ciples of  elasticity  the  maker  of  spring  scales  must  be 
familiar.  If  no  weights  are  applied  to  a  spring  scale 
greater  than  that  corresponding  to  the  elastic  limit, 
the  strain  will  always  be  proportioned  to  the  weight. 
If  then  we  mark  the  positions  of  the  pointer  when  the 
scale  pan  is  empty  and  when  it  carries  N  pounds,  we 
may  divide  the  total  deflection,  or  distance  between 
these  two  marks,  into  N  equal  spaces  and  number  the 
dividing  lines  correspondingly.  The  process  is  that 
of  calibration.  Thereafter  we  may  use  it  as  a  direct 


32  THE  REALITIES  OF  MODERN  SCIENCE 

reading  scale,  remembering  only  that  we  must  recali- 
brate if  any  doubt  ever  arises  as  to  its  accuracy. 

We  have  so  far  considered  only  standards  of  length 
and  of  weight.  Of  course,  units  of  area  and  of  volume, 
e.g.  the  square  foot  and  the  cubic  foot,  may  be  formed 
upon  the  basis  of  a  unit  of  length.  Now  it  so  happens 
that  all  the  apparently  complicated  quantities l  which 
enter  into  science,  such  as  the  heat  received  from  the 
sun,  the  intensity  of  a  sound,  the  strength  of  an  electrical 
current,  can  be  expressed  by  the  aid  of  measurements 
of  length,  weight  (or  more  strictly  mass),  and  time. 

The  idea  of  time  is  probably  one  of  the  first  abstract 
ideas  of  mankind.  We  may  think  of  time  as  flowing, 
like  an  endless  stream.  It  is  when  we  realize  that  it 
is  slipping  by  and  that  changes  are  occurring  that  we 
become  most  interested  in  how  fast  they  are  occurring. 
In  much  of  our  study  of  physical  science  and  of  its 
application  to  human  needs  we  are  interested  in  the 
time-rate  of  change,  that  is,  in  "how  fast."  So  im- 
portant is  this  idea  that  a  whole  branch  of  mathematics 
was  invented  to  deal  with  problems  involving  rates. 
This  is  known  to-day  as  the  " differential  calculus," 
although  as  originally  named  by  the  inventor  it  was 
called  " fluxions,"  that  is,  the  mathematics  which  dealt 
with  things  that  flowed. 

An  illustration  not  only  of  this  idea  of  time  as  some- 
thing which  flows  continuously  but  of  the  use  of  an 
unusual  unit  for  measuring  it,  is  found  in  the  story  of 
Galileo,  a  sixteenth-century  scientist,  and  his  study 
of  the  pendulum.  In  the  cathedral  at  Pisa,  he  noticed 

1  With  the  exception  of  two  quantities,  namely,  the  permeability 
/*,  and  the  specific  inductivity  K,  of  the  ether. 


WEIGHTS   AND   MEASURES  33 

that  the  lamps  suspended  by  long  chains  from  the  ceil- 
ing swung  to  and  fro.  The  question  entered  his  head 
as  to  whether  or  not  the  period,  i.e.  time  of  one  com- 
plete swing,  depended  upon  the  amplitude  of  the 
swing,  that  is,  upon  how  far  the  swinging  lamp  departed 
from  the  vertical. 

The  problem  then  was  to  find  out  if  the  period 
changed  as  the  motion  damped  down.  The  lengths 
of  these  pendula  were  comparatively  long,  so  that 
their  periods  were  matters  of  several  seconds.  Galileo 
timed  the  swings  by  counting  his  pulse.  In  the  interval 
between  pulse  beats  he  had  a  convenient  unit  of  time 
for  his  observations.  Although  the  pulse  rate  varies 
from  person  to  person  and  from  time  to  time  in  the 
same  person,  for  a  few  minutes,  to  determine  which 
of  two  moving  bodies  is  traveling  its  path  in  the  shorter 
time,  it  might  give  a  fair  indication. 

Up  to  this  time  pendulum  clocks  were  unknown,  and 
Galileo's  discovery  that  the  period  was  independent 
of  the  amplitude  was  the  basis  of  Huyghens'  later  use 
of  a  pendulum  to  control  the  escapement  and  hence 
the  rate  of  turning  of  the  wheels  of  a  clock.  In  a  clock 
the  springs  or  weights  turn  the  wheels  and  the  function 
of  the  pendulum  is  to  allow  only  a  definite  part  of  a 
rotation  for  each  swing,  the  amount  depending  upon 
the  relation  of  the  number  of  cogs  on  the  wheels.  The 
tendency  of  the  motion  of  the  pendulum  to  die  out  is 
cared  for  by  arranging  the  mechanism  so  that  as  it 
passes  through  the  middle  point  in  its  swing  it  receives 
a  little  kick  which  keeps  it  going  and  overcomes  the 
friction  of  the  bearing  and  of  the  air. 

The  earliest  clocks  were  somewhat  similar  in  prin- 


34  THE   REALITIES  OF   MODERN   SCIENCE 

ciple  to  the  familiar  sandglass.  They  were  large 
water  bottles,  and  the  unit  of  time  was  that  required 
for  the  water  to  run  out.  In  Greece,  where  public 
speaking  reached  perhaps  its  highest  point  in  history, 
such  clocks,  called  clepsydra,  were  used  to  time  the 
speakers.  In  one  of  the  orations  of  Demosthenes 
appear  the  words,  "you,  there,  stop  the  water." 

To-day  the  unit  of  time  which  is  commonly  adopted 
is  the  second,  being  the  sixtieth  part  of  a  minute, 
which  in  turn  is  the  sixtieth  part  of  an  hour,  which  is 
one  twenty-fourth  of  a  mean  solar  day.  Now  a  solar 
day  is  to  be  found  as  follows.  As  the  earth  revolves 
the  stars  seem  to  an  observer  to  revolve  about  an  axis 
which  is  drawn  from  the  center  of  the  earth  to  a  point 
very  close  to  the  so-called  polestar,  or  North  Star.  A 
transit  is  set  up  so  that  the  telescope  lies  in  the  plane 
formed  by  the  axis  of  the  celestial  sphere,  which  we 
have  just  described,  and  the  vertical  line  from  the  transit 
instrument  to  the  center  of  the  earth.  This  plane 
is  the  meridian  plane  at  the  location  of  the  instrument. 
At  noon  the  sun  appears  to  cross  this  meridian.  The 
time  between  two  successive  transits  of  the  sun  is  called 
a  solar  day.  Because  the  earth's  movement  about  the 
sun  is  along  an  ellipse  the  solar  day  differs  in  duration 
from  day  to  day.  Its  average  duration  is  the  funda- 
mental unit  of  time  from  which  we  obtain  the  more 
convenient  and  shorter  unit  of  the  second. 


CHAPTER  IV 
THE  BEGINNINGS  OF  SCIENCE 

SCIENCE  is  said  to  have  begun  with  Thales.  Of 
Thales  himself  we  know  but  little,  and  that  through 
writers  like  Aristotle  who  lived  two  hundred  years 
later.  Specifically,  he  is  credited  by  the  latter  with 
accounting  for  the  attraction  of  iron  by  a  lodestone,  by 
assuming  a  cause  inherent  in  the  stone  instead  of  super- 
natural influences,  as  did  his  contemporaries.  With 
him  our  knowledge  of  magnetism  begins. 

Thales,  also,  felt  the  need  of  accounting  for  the 
various  kinds  of  matter  in  terms  of  some  common 
element.  Twenty-four  centuries  passed  between  his 
suggestion  and  the  common  acceptance  of  the  atomic 
theory  which  describes  all  matter  in  terms  of  a  compara- 
tively few  elements  in  various  proportion.  But  Thales 
envisaged  a  single  element.  This  accords  with  the 
extension  of  the  atomic  theory,  the  "  electron  theory," 
which  is  accepted  to-day.  Atoms  of  all  substances 
are  now  known  to  be  composed  of  small  particles  of 
electricity,  called  electrons,  which  are  all  alike  without 
regard  to  the  chemical  element  from  which  they  may 
be  obtained. 

All  Thales  really  did  was  to  direct  men's  attention 
to  the  problem,  still  incompletely  solved,  of  the  com- 
position of  matter.  In  the  centuries  since  his  time 
and  more  especially  in  the  last,  progress  has  been  made 

35 


36  THE   REALITIES  OF   MODERN   SCIENCE 

toward  a  solution.  The  amazing  productions  of  modern 
chemistry1  have  been  made  possible  by  the  knowl- 
edge that  any  homogeneous2  substance  is  composed 
of  small  particles  called  molecules  which  are  all  alike 
in  construction.  The  molecules  themselves  are  formed 
by  the  combination  of  still  smaller  particles,  the  atoms, 
which  were  once  thought  to  be  indivisible.  Of  the 
atoms  there  are  some  ninety  different  kinds,  each  a 
chemical  element.  With  the  ways  in  which  they  com- 
bine, the  properties  of  their  compounds,  and  the 
methods  by  which  elements  may  be  extracted  from 
compounds,  modern  chemistry  is  concerned. 

Now  Thales  advanced  a  theory  as  to  the  composi- 
tion of  matter.  Of  course,  his  theory  was  all  wrong 
because  he  had  insufficient  experimental  data.  He 
suggested  that  water  was  the  single  fundamental 
element.  The  progress  of  science  is  marked  by  theories 
which  have  been  advanced  by  various  scientists,  ac- 
cepted and  held  for  a  time  by  other  scientists,  and  later 
superseded.  Let  us  see  what  a  scientist  means  when  he 
speaks  of  a  theory.  He  should  mean  that  upon  the 
assumption  of  certain  things  as  true  he  can  then  ex- 
plain in  terms  of  these  all  the  known  facts  of  the  subject 
of  his  theory.  He  does  not  mean  that  these  assump- 
tions are  necessarily  correct.  If  they  help  to  form  a 
picture  of  the  mechanism  of  known  phenomena  they 
may  enable  him  to  predict  what  will  happen  under 
somewhat  different  conditions. 

1  Cf.  Duncan,  "  Some  Chemical  Problems  of  To-day."     Harpers, 
1911. 

2  The  word  "homogeneous"  is  also  applied  to  substances  which 
are  mixtures  in  definite  proportions  of  two  or  more  kinds  of  mole- 
cules. 


THE   BEGINNINGS  OF  SCIENCE  37 

If  under  new  conditions  what  he  predicts  from  his 
theory  fails  to  happen,  he  knows  that  it  is  wrong  some- 
where. He  therefore  modifies  it  by  changing  some 
assumptions  or  adding  new  factors  until  he  can  account 
for  the  phenomenon  which  invalidated  the  theory. 
It  may  be  that  the  theory  is  shown  to  be  absolutely 
wrong,  so  that  no  modification  can  possibly  make  it 
explain  the  things  that  actually  happened.  In  that 
case  it  must  be  discarded  at  once. 

As  time  goes  on  new  conditions  arise  and  the  theory 
is  subjected  to  new  tests.  Sometimes  it  is  not  neces- 
sary to  wait  until  new  conditions  arise  naturally,  for 
they  may  be  devised  and  arranged.  A  scientist  thus 
checks  his  theory  by  experiment.  It  may  be  that  he 
can  form  a  rule,  expressing  what  to  expect  under  any 
given  set  of  conditions.  This  relation  between  cause 
and  effect  he  states  as  a  law.  Sometimes  the  experi- 
ments will  show  unexpected  relations,  not  inconsist- 
ent with  the  theory  and  yet  not  covered  by  it.  The 
scientist  may  then  verify  this  relation  in  other  experi- 
ments and  make  a  statement  of  it  as  a  purely  empirical 
law. 

By  theories  formulated  on  the  basis  of  experiment, 
by  experiments  devised  to  test  them,  and  by  their 
consequent  verification,  modification,  or  rejection, 
science  has  advanced.  It  has  developed  through  a 
succession  of  theories.  The  man  who  furthers  it, 
except  by  some  accidental  discovery,  is  the  one  who  has 
imagination  for  the  hidden  processes  by  which  natural 
phenomena  occur. 

The  danger  to  science  is  not  in  the  suggestion  of  new 
theories  but  in  the  failure  of  their  adherents  to  follow 


38  THE  REALITIES   OF   MODERN  SCIENCE 

the  steps  traced  above.  It  sometimes  happens  that  a 
theory  is  apparently  completely  verified  and  for  genera- 
tions commonly  accepted  by  the  scientifically  trained. 
Later  some  experimental  evidence  may  come  to  notice 
which  explodes  the  existing  theory.  The  new  theory 
which  arises  to  take  into  account  the  exceptions  to  the 
previously  accepted  theory  must,  however,  fight  its 
way  against  a  natural  conservatism  and  even  against  the 
opinions  of  scientists  whose  other  achievements  have 
entitled  them  to  great  influence.1 

In  the  case  of  the  Greek  scientists  or  rather  philoso- 
phers, the  tendency  was  to  advance  broad  general 
theories,  like  that  of  Thales,  without  applying  to  them 
the  check  of  experiment.  Of  course,  if  by  experiment 
we  can  learn  how  some  natural  phenomenon  occurs, 
we  no  longer  need  a  " theory"  to  account  for  it,  nor 
need  we  build  our  explanation  upon  a  "  hypo  thesis." 
On  the  other  hand,  checking  a  theory  by  experiment 
does  not  indicate  the  correctness  of  all  the  assumptions 
but  merely  that  they  will  account  for  all  the  observed 
facts  of  the  experiment. 

Forming  theories  is  then  one  of  our  methods -of 
learning  the  explanation  of  the  phenomena  which 
attract  our  attention.  Without  curiosity  as  to  the 
"why"  of  the  various  aspects  of  the  world  about  us 
we  make  but  slight  progress.  The  Greeks  had  an 
unflinching  curiosity.  Although  most  of  their  physical 
theories  have  since  proved  wrong,  the  curiosity  which 

1  In  the  struggle  between  the  wave  theory  of  light  and  the  pre- 
ceding corpuscular  theory  the  opinion  of  Newton  greatly  retarded 
the  common  acceptance  of  the  former.  The  classical  example  is, 
of  course,  the  astronomical  theory  of  Ptolemy. 


THE   BEGINNINGS  OF  SCIENCE  39 

urged  their  formulation  initiated  scientific  investigation. 
It  is  not  the  contributions  to  the  total  of  scientific  facts 
which  distinguishes  them  from  the  Babylonians  and 
Egyptians  but  their  contribution  of  method  and  of 
point  of  view. 

We  all  recognize  how  necessary  to  success  are  a 
proper  point  of  view  and  a  good  method.  The  selec- 
tion of  the  best  method  has  led,  in  manufacturing,  to 
motion  studies  and  the  standardization  alike  of  work- 
men and  mechanical  parts.  This  whole  matter  of 
efficiency  1  is  a  fascinating  and  important  study  which 
concerns  us  all.  Of  course,  a  proper  method  of  study 
is  as  important  to  the  student  as  is  an  efficient  method 
to  the  worker.  To  some  extent  the  arrangement  and 
facilities  of  a  school  and  the  formal  instruction  of 
teacher  and  textbook  help  the  student.  Very  largely, 
however,  it  depends  on  the  pupil,  on  his  own  critical  atti- 
tude toward  his  mental  processes,  whether  or  not  he  ever 
forms  an  efficient  method  of  studying,  that  is,  of  grasp- 
ing and  making  his  own  new  ideas.  The  student  of 
science  who  fails  to  acquire  the  point  of  view  of  the 
physicist  and  a  sympathy  with  his  methods  may  be 
stuffed  as  full  of  facts  as  an  encyclopedia  and  yet,  like 
the  latter,  be  unable  to  develop  an  original  idea  or  to 
solve  a  new  problem. 

To  the  Greeks  natural  phenomena  were  question 
marks,  stimulating  their  curiosity  and  demanding 
answers.  Before  the  days  of  their  philosophers,  physi- 
cal phenomena  were  explained  in  supernatural  terms 

1  How  best  to  employ  such  methods  of  efficiency  without  in- 
hibiting the  creative  instinct  is  an  interesting  question.  Cf.  Helen 
Marot,  "The  Creative  Instinct  in  Industry." 


40  THE   REALITIES   OF   MODERN   SCIENCE 

as  being  the  work  of  some  spirit  or  deity.  Thus 
on  the  island  of  Samothrace  there  was  a  "mystery," 
in  charge  of  the  priests,  which  attracted  travelers  from 
large  distances.  From  the  accounts  it  seems  to  have 
consisted  of  a  lodestone  and  some  iron  rings  which 
were  attracted  to  this  natural  magnet  and  hung  sus- 
pended from  it.  Thales  got  far  enough  scientifically 
to  assign  this  attraction  to  something  inherent  in  the 
stone  itself  rather  than  to  any  action  of  spirits.  As  a 
matter  of  fact,  however,  the  simple  laws  of  magnetism 
were  not  accurately  stated  until  they  were  derived 
by  Gilbert  (1600)  from  long  and  careful  experi- 
ment. The  fact  that  the  curiosity  of  the  Greek  phi- 
losophers did  not  falter  before  what  was  commonly  con- 
sidered supernatural  is,  however,  illustrated  by  this 
case  of  Thales. 

Of  course,  to-day  we  should  not  be  prevented  by 
superstitions l  from  seeking  the  explanation  of  any 
phenomena  which  may  attract  our  attention.  But 
there  was  more  to  the  Greek  attitude  than  mere  curi- 
osity. They  observed  and  classified  phenomena.  This 
correlation  of  phenomena,  that  is,  the  collection  of 
those  which  are  similar  in  principle,  their  description 
in  abstract  terms,  the  generalization  of  the  underlying 
laws,  and  the  formulation  of  theories,  constitutes 
science.  The  influence  of  the  Greek  is  evident  to-day 
even  in  the  words  we  use ;  for  example,  our  word 
"principle"  conveys  an  idea  first  expressed  by  Aris- 
totle (384-322  B.C.). 

The  purpose  of  Aristotle  illustrates  the  Greek  point 

1  Many,  however,  have  inhibitions  against  certain  methods  and 
fields  of  research,  e.g.  psychoanalysis. 


THE  BEGINNINGS  OF  SCIENCE  41 

of  view.  It  was  "to  comprehend,  to  define  and  to 
classify  the  phenomena  of  organic  and  inorganic  nature, 
to  systematize  the  knowledge  of  his  own  time.  He 
pressed  his  way  through  the  mass  of  things  knowable 
and  subjected  its  diversity  to  the  power  of  his  own 
thought.  No  wonder  that  for  ages  he  was  known  as 
'The  Philosopher/  that  is,  master  of  those  who  know. 
His  great  systematizing  intellect  has  left  its  impress  on 
nearly  every  department  of  human  knowledge.  Physi- 
cal astronomy,  physical  geography,  meteorology,  phys- 
ics, chemistry,  geology,  botany,  anatomy,  physiology, 
embryology,  and  zoology  were  enriched  by  his  teach- 
ing. It  was  through  him  that  logic,  ethics,  psychology, 
rhetoric,  aesthetics,  political  science,  zoology  (especially 
ichthyology)  first  received  systematic  treatment." 

These  Greeks  were  keen  observers.  Aristotle  asks 
such  questions  as  "Why  are  vehicles  with  large  wheels 
easier  to  move  than  those  with  small?"  and  "Why 
do  objects  in  a  whirlpool  move  toward  the  center?" 
He  also  reasoned  that  the  earth  is  spherical  because 
during  an  eclipse  of  the  moon  by  the  earth  the  edge 
of  its  shadow  on  the  moon  is  an  arc.  He  said  that  it 
was  therefore  possible  "that  the  region  about  the  Pillars 
of  Hercules  is  connected  with  that  of  India,  and  that 
there  is  thus  only  one  ocean."  This  idea,  transmitted 
to  Columbus  through  the  writings  of  Roger  Bacon, 
a  Franciscan  monk  (1214-1294),  influenced  him  to  at- 
tempt his  first  voyage  of  discovery. 

Many  other  Greeks  of  this  golden  age,  or  somewhat 
earlier,  also  contributed  to  the  method  and  to  the 
facts  of  science.  There  is  the  principle  of  Archimedes 
that  bodies  immersed  in  a  fluid  are  buoyed  up  with  a 


42  THE   REALITIES   OF   MODERN  SCIENCE 

force  equal  to  the  weight  of  the  fluid  they  displace. 
Of  course,  if  a  body  is  not  entirely  immersed  it  is 
buoyed  up  with  a  force  equal  to  the  weight  of  the  fluid 
which  the  immersed  portion  of  the  body  displaces.  We 
speak  of  ships  to-day  in  tons  displacement,  stating 
the  weight  of  the  water  displaced  and  hence  a  measure 
of  the  total  weight  which  the  ship  and  cargo  may  have 
if  its  tendency  to  sink  is  to  be  neutralized  by  the 
buoyancy  of  the  water. 

The  story  of  how  Archimedes  (287-212  B.C.)  hap- 
pened to  arrive  at  his  law  is  told  by  Vitruvius,  the 
Roman  engineer  and  architect  who  wrote  about  the 
time  of  Christ.  "  Though  Archimedes  discovered 
many  curious  matters  that  evidenced  great  intelli- 
gence, that  which  I  am  about  to  mention  is  the  most 
extraordinary.  Hiero,  when  he  obtained  the  regal 
power  of  Syracuse,  having,  on  the  fortunate  turn  of 
his  affairs,  decreed  to  be  placed  in  a  certain  temple  a 
votive  crown  of  gold  to  the  immortal  gods,  commanded 
it  to  be  made  of  great  value,  and  assigned  to  the  manu- 
facturer for  this  purpose  an  appropriate  weight  of 
the  metal.  The  latter  in  due  time  presented  the  work 
to  the  king.  It  was  beautifully  wrought  and  the  weight 
appeared  to  correspond  to  that  of  the  gold  which  had 
been  assigned  for  it.  But  a  report  was  circulated  that 
some  of  the  gold  had  been  abstracted  and  that  the  de- 
ficiency thus  caused  had  been  supplied  by  silver.  Hiero 
was  indignant  at  the  fraud,  and  unacquainted  with  a 
method  by  which  the  theft  might  be  detected,  requested 
Archimedes  to  give  it  his  attention.  While  charged 
with  this  commission  Archimedes  went  by  chance 
to  a  bath.  On  getting  into  the  water  he  noticed  that 


THE   BEGINNINGS  OF  SCIENCE  43 

just  in  proportion  as  his  body  was  immersed  did  the 
water  run  out  of  the  tub.  From  this  he  caught  the 
method  to  be  followed  in  the  solution  of  his  problem 
and  immediately  followed  it  up.  He  jumped  out  of 
the  tub  in  his  joy  and  ran  home  naked  through  the 
streets,  crying  in  a  loud  voice  ' Eureka/  meaning  'I 
have  found  it,'  for  he  had  found  the  method1  for 
which  he  was  searching." 

Another  story  as  to  Archimedes  is  told  by  Plutarch. 
While  conversing  with  Hiero,  Archimedes  explained 
the  principle  of  the  lever  by  an  illustration.  He  said 
that,  given  a  lever  and  another  world  like  our  earth  on 
which  to  stand,  he  could  move  our  earth  itself.  Hiero 
was  struck  with  amazement,  and  asked  Archimedes 
to  give  a  demonstration  by  moving  some  large  weight 
on  this  earth.  The  latter  arranged  to  draw  out  of  the 
dock  of  the  king's  arsenal  a  heavily  loaded  ship  which 
was  otherwise  only  to  be  moved  by  many  men  with 
great  effort.  To  accomplish  this  he  used  a  system  of 
pulleys  and  unaided  drew  the  boat  out  of  its  dock. 

How  this  "  technical  Yankee  of  antiquity,"  as  he 
has  been  called  by  an  eminent  German  historian, 

1  Suppose  Archimedes  weighed  the  crown  first  in  air  and  then  in 
water.  When  hung  from  the  balance  arm  so  as  to  be  immersed 
in  water  it  weighed  less  than  before  by  an  amount  equal  to  that  of 
the  displaced  water.  Hence  subtracting  the  two  gave  the  weight  of 
the  displaced  water.  Now  suppose  he  performed  the  same  oper- 
ations on  a  lump  of  gold.  He  would  have  found  that  the  weight 
of  the  gold  in  air  was  19.3  times  the  weight  of  the  displaced  water. 
If  he  used  a  lump  of  silver  he  found  this  ratio  to  be  only  10.5.  Hence 
if  the  crown  was  pure  gold  it  should  weigh  19.3  times  as  much  in 
air  as  the  water  it  displaced.  On  the  other  hand,  if  it  was  of  pure 
silver  this  ratio  would  be  10.5.  If  it  was  a  mixture  of  the  two 
metals  the  ratio  would  lie  between  these  two  values. 


44  THE   REALITIES  OF   MODERN   SCIENCE 

developed  a  water  screw  for  pumping,  which  was  used 
in  Egypt  for  irrigation ;  how  he  invented  machines, 
like  catapults,  for  hurling  huge  stones,  which  were 
used  in  the  defense  of  Syracuse  against  the  Romans  in 
212  B.C.  ;  and  how,  when  the  Romans  finally  captured 
the  city,  he  was  killed  by  a  Roman  soldier  whom  he 
urged  not  to  spoil  the  circles  of  the  geometrical  problem 
which  he  was  diagraming  on  the  sand,  are  interest- 
ing stories  which  the  reader  may  find  recorded  in  his- 
tories of  science. 

How  Euclid,  Archimedes,  and  other  of  these  Greek 
philosophers  developed  geometry;  how,  for  example, 
Archimedes  arrived  at  the  value  of  pi,  TT,  the  ratio  of 
the  circumference  of  a  circle  to  its  diameter,  as  being 
something  between  3y  and  Sfr,  and  of  their  astro- 
nomical theories,  the  student  will  also  find  interesting 
reading.1 

Despite,  however,  the  many  contributions  of  the 
Greeks  to  science,  they  frequently  relied  too  much  on 
their  mental  processes  and  not  enough  on  experiment. 
The  phenomena  of  nature  had  not  at  that  time  been 
sufficiently 2  observed  for  correlation.  They  usually 
had  too  meager  data  on  which  to  base  conclusions. 
Of  similar  errors  the  popular  science  of  to-day  is  oc- 
casionally guilty.  The  Greeks  also  made  a  distinc- 
tion between  theory  and  practice  which  had  some  re- 

1  Cf .  Tyler  and  Sedgwick,  "A  Short  History  of  Science."     The 
Macmillan  Company,  1917. 

2  But  some  Greeks  recognized  this,  as  is  seen  in  a  statement  of 
Aristotle  that ' '  the  phenomena  are  not  yet  sufficiently  investigated ; 
when  they  are,  then  one  must  trust  more  to  observation  than  to 
speculation,  and  to  the  latter  no  further  than  it  agrees  with  the 
phenomena." 


THE   BEGINNINGS  OF  SCIENCE  45 

tarding  effect  on  science  in  later  ages,  for  the  writings 
of  these  philosophers,  particularly  of  Aristotle,  were 
the  textbooks  of  the  middle  ages.1 

Our  words  " theory "  and  " practice"  are  derived 
from  words  introduced  by  them.  These  words  were 
used  to  represent  two  opposite  ideas,  and  this  distinc- 
tion and  separation  is  the  real  fallacy  in  their  philosophy 
as  far  as  concerns  natural  science.  We  use  the  words 
in  essentially  the  same  way  to-day,  when  we  are  speak- 
ing carelessly ;  thus  we  say,  "  Oh,  he  doesn't  know  why 
it  works;  he  is  just  a  practical  man,"  or  "He  may 
know  it  theoretically,  but  he  is  very  impractical." 
To  the  Greeks,  theory  was  rational  and  pure,  while 
practice  was  irrational  and  base.  One  dealt  with 
celestial  matters  while  the  other  dealt  with  terrestrial. 
Theory  was  noble  and  practice  was  ignoble.  This 
latter  distinction  was  probably  due  largely  to  the  exist- 
ence of  the  institution  of  human  slavery.  The  me- 
chanical tasks,  like  mining  and  later  agriculture,  were 
largely  performed  by  slave  labor.  This  distinction 
between  noble  and  servile  persisted  even  after  slavery 
had  been  replaced  by  feudalism,  and  persists  to-day. 

It  was  natural  that,  with  this  differentiation  in  mind, 
the  Greeks,  and  those  schoolmen  of  the  middle  ages 
who  followed  their  methods,  should  rely  too  much  on 
deductive  reasoning,  proceeding  from  broad  and  general 
premises  to  determine  what  should  be  true  in  specific 
cases.  In  the  hands  of  the  schoolmen  science  became 
a  matter  of  a  priori  truths,  independent  of  experience 
and  experiment. 

1  From  the  decline  of  the  Roman  Empire  to  the  Renaissance 
(about  300  to  1400  A.D.). 


46  THE  REALITIES  OF   MODERN  SCIENCE 

An  illustration  is  their  reasoning  as  to  the  speed  of 
falling  bodies.  They  had  probably  noticed  that  a 
pebble  falls  faster  than  a  feather,  and  they  concluded 
that  "  bodies  fall  with  velocities  proportional  to  their 
weights."  This  conclusion  was  not  tested  by  observing 
the  fall  of  two  bodies  of  different  weights,  but  with 
essentially  the  same  air  resistance,  until  Galileo  in 
1590  actually  tried  the  experiment.  From  the  leaning 
tower  of  Pisa,  a  height  of  about  180  feet,  he  dropped 
balls  of  different  materials  and  of  different  sizes. 
He  found  that  they  fell  in  almost  the  same  times,  and 
that  light  objects  like  pieces  of  paper  fell  more  nearly 
like  heavy  balls  when  the  paper  was  tightly  wadded  into 
a  ball.  Reasoning  a  posteriori  he  concluded  that  except 
for  the  resistance  of  the  air  all  bodies  would  fall  through 
the  same  height  in  the  same  time.  Galileo  could  not 
perform  the  experiment  under  conditions  where  air 
friction  and  buoyancy  were  eliminated,  for  the  air  pump 
was  not  invented  until  about  sixty  years  later.  But 
since  he  found  that  the  more  nearly  the  effects  of  the 
air  on  two  falling  bodies  were  equalized,  the  more 
nearly  did  they  fall  through  the  same  height  in  equal 
times,  he  reasoned  for  the  limiting  case  that  the  two 
bodies  would  fall  with  equal  velocities. 

The  importance  in  science  of  inductive  reasoning 
was  not  fully  appreciated  until  even  after  Galileo's 
time,  although  from  the  time  of  Roger  Bacon  there 
were  men  who  advocated  it.  With  the  definitions  of 
these  two  forms  of  reasoning  in  mind  we  may  now  see 
more  clearly  the  method  of  science.  From  a  mass  of 
correlated  information  as  to  some  class  of  natural 
phenomena,  scientists  reason  inductively  to  obtain  a 


THE   BEGINNINGS  OF  SCIENCE  47 

general  law  for  the  relations  which  appear  to  exist  in 
these  phenomena. 

Gilbert's  experiments  are  an  early  instance  of  this 
method.  He  formed  a  small  sphere  of  a  magnetite 
(lodestone)  and  observed  the  behavior  of  small  pieces 
of  iron  placed  on  the  sphere.  He  found  that  a  bit  of 
iron  tended  to  lie  along  a  meridian  line  of  this  "ter- 
rella"  just  as  does  a  compass  needle  on  the  surface  of 
the  earth.  From  these  and  other  experiments  he  came 
to  the  conclusion  that  the  terrestrial  globe  is  itself  a 
magnet.  He  found  that  the  same  end  or  pole  always 
pointed  to  the  same  pole  of  the  earth  and  hence  that 
there  was  in  each  magnet  a  north  seeking  pole  and  a 
south  seeking  pole.  By  experiments  with  two  magnets 
he  found  that  their  north  seeking  poles  when  placed  in 
proximity  are  urged  away  from  each  other ;  similarly 
as  to  their  south  seeking  poles.  But  a  north  seeking 
pole  and  the  south  seeking  pole  of  the  other  magnet 
are  urged  toward  each  other.  From  these  experiments 
he  reached  inductively  the  general  law  of  the  action 
of  magnets,  that  like  poles  repel  and  unlike  poles 
attract.  To-day,  there  are  other  ways  than  those 
known  to  Gilbert  for  obtaining  a  magnetic  effect ;  but 
no  matter  how  it  is  produced,  we  may  reason  deduc- 
tively from  his  general  law. 

The  methods  of  science  are  thus  seen  to  be  both 
inductive  and  deductive.  The  former  lead  to  exten- 
sions of  the  boundaries,  the  latter  to  extensions  of 
application. 


CHAPTER  V 

THE  BEGINNINGS   OF  EXPERIMENTATION 

Two  thousand  years  elapsed  between  the  beginnings 
which  the  early  Greeks  made  in  the  correlation  of 
knowledge  and  the  initiation  of  logical  experimenta- 
tion, which  has  made  possible  the  science  of  to-day. 
Observation,  and  correlation  based  upon  it,  had  in- 
deed been  considerable  before  the  time  of  Gilbert  and 
Galileo,  whom  we  consider  as  the  first  experimenters 
in  physics.  In  astronomy,  experiments  are  impossible 
and  progress  is  made  from  the  classified  data  of  many 
observations.  The  laws  of  other  physical  science  have 
been  obtained  largely  as  the  result  of  the  observation 
of  the  phenomena  taking  place  under  conditions  arti- 
ficially prepared  to  facilitate  such  observations. 

What  civilizations  followed  the  Greek  and  what 
were  the  peculiar  characteristics  of  their  bearers  to 
make  these  years  so  unproductive  ?  The  widest-spread 
empire  of  the  ancient  world  was,  of  course,  the  Roman, 
which  by  the  time  of  Julius  Caesar  had  practically 
covered  the  then  known  world.  But  the  genius  of 
the  Romans  was  essentially  military  and  not  scientific. 
Their  achievements  in  thought  were  largely  in  the 
science  of  government.  In  the  method  of  organizing 
their  growing  domains,  in  the  enactment  and  codifica- 
tion of  the  necessary  laws,  and  in  the  general  develop- 
ment of  legal  procedure,  they  made  their  chief  contri- 

48 


THE   BEGINNINGS  OF  EXPERIMENTATION       49 

butions  to  human  knowledge.  Their  national  spirit 
showed  itself  in  the  conquest  and  consolidation  of 
alien  territory.  What  developments  they  made  in 
natural  science  were  dictated  by  this  spirit  and  its 
necessities,  and  were  largely  in  the  field  of  military 
engineering. 

In  this  field  they  surpassed  previous  civilizations. 
To  maintain  their  lines  of  military  communication 
they  made  advances  in  the  science  of  road  building  and 
bridge  making.  The  etymology  of  the  word  "ponti- 
fex,"  which  was  the  name  of  the  highest  priesthood,  is 
immediately  suggestive.  Their  aqueducts  and  public 
buildings  are  justly  famous.  Agriculture  and  forestry 
were  held  in  good  repute ;  and  in  early  Rome  the  pa- 
trician was  always  a  farmer.  The  work  of  the  arts  was 
not  highly  regarded,  even  before  the  enormous  growth 
of  slavery  had  attached  its  stigma  to  almost  all  pro- 
ductive work. 

In  medicine,  or  more  particularly  in  surgery,  the 
Romans  made  contributions.  In  the  ruins  of  Pom- 
peii, which  was  destroyed  by  Vesuvius  in  79  A. D.,  many 
interesting  surgical  instruments  have  been  found.  It 
was  this  eruption  of  the  volcano  of  which  Pliny,  the 
Elder  (23-79  A.D.),  in  scientific  curiosity,  took  too 
close  a  view.  Pliny  himself  wrote  a  long  book  on 
" Natural  History"  which  illustrates  the  practicality 
of  the  Roman  interest  in  science. 

How  the  Roman  Empire  gradually  succumbed  to 
the  waves  of  invasion  of  rude  strong  tribes  from  north- 
eastern Europe  and  how  "dark  ages"  followed  is  one 
of  the  important  stories  of  "medieval  history."  How 
all  branches  of  learning  and  all  forms  of  art  suffered 


50  THE   REALITIES  OF   MODERN  SCIENCE 

a  practical  death  or  cessation  in  these  centuries,  how 
between  1200  and  1600  they  were  revived,  our  histories 
tell  us.  But  to  the  scientist  there  may  well  be  par- 
ticular interest  in  the  events  by  which  science  was 
preserved  and  even  fostered  in  its  growth  during  these 
centuries. 

In  our  Arabic  numerals  we  have  a  trace  and  an  ex- 
ample of  the  contributions 1  of  non-European  races 
during  the  medieval  ages.  Through  their  capture  of 
Alexandria  in  641  A.D.  the  Arabs  had  become  the 
custodians  of  the  records  of  Greek  learning.  This 
city,  founded  at  the  mouth  of  the  Nile  by  Alexander 
the  Great  in  about  330  B.C.,  was  the  home  of  many 
famous  scientists  and  mathematicians,  whom  it  at- 
tracted by  its  museum  and  its  libraries.  Here  lived 
Euclid  (300  B.C.).  Here  also,  about  150  A.D.,  the  as- 
tronomer Ptolemy  taught  his  theory  of  a  geocentric 
celestial  system,  which  was  accepted  until  Copernicus 
(1473-1543)  demonstrated  that  the  sun  was  the  center. 
About  the  same  time  Alexandria  also  produced  the 
noted  inventor  Hiero,  whose  steam  engine  is  a  sort 
of  anticipation  of  the  modern  steam  turbine.  To 
the  records  of  these  and  other  achievements,  as  well 
as  to  the  entire  field  of  Greek  thought,  the  Arabs  were 
appreciative  heirs,  while  Europe  suffered  a  decay  of 
learning. 

In  the  meantime  Christianity  had  spread  widely 
throughout  Europe.  The  bishops  of  its  various  centers 
had  become  responsible  to  the  bishop  at  Rome,  who 
thus  became  the  Pope.  It  had  been  accepted  by  the 

1  Algebra  and  trigonometry  are  of  Arabic  or  at  least  of  non-Eu- 
ropean origin. 


THE  BEGINNINGS  OF  EXPERIMENTATION       51 

invading  barbarians  in  whole  armies.  By  the  time  of 
Charlemagne  in  800  the  union  of  Church  and  State 
had  been  formed  which  was  to  dictate  the  life  of  Europe 
for  centuries. 

This  dictation  entered  into  science  as  well  as  into 
religion.  Of  this  the  striking  example  is  the  as- 
tronomical system  of  Ptolemy.  It  is  the  easiest  of 
which  to  persuade  the  ignorant,  for  it  is  the  most 
obvious.  In  addition  it  had  been  elaborately  and  in- 
geniously supported  by  geometrical  reasoning.  How 
with  the  years  it  gathered  authority,  and  even  became 
so  firmly  held  that  disbelief  of  it  was  punished  by  the 
Christian  Church,  is  an  interesting  illustration  of 
Europe's  attitude  toward  science  in  the  medieval 
ages.1  How  it  was  ultimately  discarded  in  favor  of 
the  heliocentric  system  of  Copernicus  is  one  of  the 
important  stories  of  the  Renaissance. 

During  the  14th  and  15th  centuries  occurred  those 
marked  changes  in  the  intellectual  life  of  Europe  that 
constitute  the  Renaissance.  To  some  extent  this  was 
merely  a  revival  of  interest  in  the  learning,  that  is,  the 
literature,  art,  and  philosophy,  of  the  classical  world. 
For  centuries  this  pagan  learning  had  for  the  most 
part  been  under  the  disapproval  of  the  Christian 
Church,  although  some  of  it,  e.g.  the  work  of  Aris- 
totle, formed  part  of  the  education  of  the  time.  Such 
knowledge  as  medieval  scholars  had  of  the  courageous 
intellectual  life  of  the  Greeks  was  frequently  obtained 
from  Latin  translations,  made  by  Jews,  of  Arabic 

1  Galileo  in  his  last  years  was  compelled  to  recant  and  deny  the 
Copernican  theories  with  which  he  had  previously  expressed  his 
agreement. 


52  THE  REALITIES  OF  MODERN  SCIENCE 

commentaries  on  Greek  texts.  In  the  spirit  of  the  Re- 
naissance, men  went  directly  to  the  Greek  manuscript 
and  acquired  something  of  the  author's  passion  for 
knowledge  and  faith  in  reason.  The  invention  about 
1450  of  the  process  of  printing  from  movable  type 
enormously  increased  the  dissemination  of  this  knowl- 
edge. 

By  the  time  of  the  Renaissance  two  important  in- 
ventions of  the  Chinese  had  reached  Europe.  These 
were  gunpowder  and  the  magnetic  compass.  With 
the  latter  as  a  guide,  Portuguese  sailors  had  penetrated 
into  strange  water  and  explored  long  stretches  of  the 
African  coast.  They  had  sailed  through  the  weedy 
terrors  of  the  Sargasso  Sea  and  rounded  Cape  Bojador. 
The  old  superstitions  of  a  region  of  fire,  and  of  gales 
that  always  blew  the  sailor  away  from  home  (the  trade 
winds,  probably),  were  yielding  to  exploration.  Under 
such  conditions  there  came  to  Columbus  the  opinion 
of  Aristotle,  that  India  might  be  reached  by  sailing 
westward. 

This  case  of  Columbus  forms  an  interesting  illus- 
tration of  the  cumulative  effect  of  knowledge.  The 
revival  of  learning,  the  growing  familiarity  with  the 
compass,  the  invention  of  printing,  the  explorations 
of  other  navigators,  all  prepared  his  way.  He,  in 
his  turn,  by  his  discoveries,  pushed  further  out  the 
boundaries  of  human  knowledge.  The  more  persons 
there  are  who  know  a  given  group  of  facts  or  theories 
the  greater  is  the  chance  that  one  of  the  many  may 
make  the  extension,  to  the  possibility  of  which  the 
rest  are  blind.  Knowledge  is  more  than  power ;  it  is 
the  condition  for  growth. 


THE   BEGINNINGS  OF  EXPERIMENTATION        53 

The  first  voyage  of  Columbus  was  an  experiment. 
It  is  not  always  that  experiments  may  be  performed  in 
the  laboratory,  for  many  of  them  must  be  on  a  national 
or  world-wide  scale,  as  is  true  to-day  of  many  social 
experiments.  Sometimes  the  by-products  of  an  ex- 
periment are  themselves  of  great  value.  An  illustra- 
tion of  this  is  Columbus's  discovery  of  the  variation 
of  the  magnetic  declination.1  Not  always,  of  course, 
are  the  by-products  of  the  main  experiment  so  evident 
as  they  were  in  this  case  of  Columbus's  first  voyage ; 
nor  are  they  so  overshadowed  by  the  importance  of 
the  main  result.  In  such  by-products  may  be  the 
greatest  value  of  the  experiment.  While  the  result 
of  the  main  experiment  is  usually  the  reward  of  clear 
thinking  and  precise  manipulation  of  the  experimental 
apparatus,  the  by-products  are  usually  the  reward 
of  a  comprehending  observation.  They  are  frequently 
the  basis  of  important  inventions,  but  they  are  not 
obtained  by  the  unobserving,  who  pass  over  their 
indications. 

Three  qualifications  for  an  experimenter  are  thus 
suggested :  (1)  clear  thinking,  (2)  accurate  experi- 
mentation, and  (3)  careful  observation.  The  first  is 
required  before  the  experiment,  in  order  that  experi- 
mental conditions  may  be  devised  which  will  permit 
of  a  definite  answer  to  the  question  under  examina- 
tion. The  second  demands  that  the  measurements  be 
sufficiently  precise  not  to  obscure  the  desired  results. 
There  must  also  be  a  careful  observation  during  the 

1  The  angle  between  the  direction  assumed  by  a  compass  needle 
and  the  geographical  meridian  is  called  the  magnetic  declination  of 
a  locality. 


54  THE   REALITIES  OF   MODERN   SCIENCE 

experiment  and  close  scrutiny  of  the  data  obtained,  so 
that  new  or  unexpected  phenomena  may  be  detected. 
To-day  it  is  possible,  as  a  rule,  to  obtain  from  in- 
strument makers  apparatus  with  which  very  precise 
measurements  may  be  made.  The  average  experi- 
menter does  not  have  much  difficulty  in  satisfying 
the  second  qualification.  With  the  aid  of  accurate 
instruments  and  machines  much  of  the  experimental 
work  of  the  commercial  world  of  industry  may  be 
performed  by  assistants  who  lack  the  other  two  quali- 
fications. The  experiments  are,  however,  planned  and 
scrutinized  by  scientists  who  satisfy  more  fully  these 
other  conditions.  In  fact,  without  a  large  measure  of 
ability  of  this  character,  the  experimenter  becomes  but 
a  human  part  of  the  machine  which  he  operates.  An 
experimenter  who  lacks  the  first  and  third  qualifica- 
tions bears  to  a  true  scientist  the  same  relation  as  the 
driver  of  a  locomotive  which  is  guided  by  the  rails  and 
switches  bears  to  those  navigators  like  Columbus  who 
sailed  across  uncharted  seas. 

Of  the  true  scientist  the  seventeenth  century  furnishes 
many  good  examples.  Of  Gilbert  and  Galileo  we  have 
already  learned.  Of  Torricelli  (1608-1647),  Pascal 
(1623-1662),  Boyle  (1627-1691),  Hooke  (1635-1703), 
and  Newton  (1642-1727)  it  is  desirable  to  learn  more, 
since  various  principles  or  laws  of  physics  are  known 
by  the  names  of  these  discoverers.  For  example, 
the  statement  that  stress  is  proportional  to  strain,  is 
Hooke's  Law. 

Torricelli  was  a  disciple  of  Galileo  who  devised  the 
barometer.  Now,  the  ancients  had  used  pumps  for 
lifting  water,  and  had  observed  that  in  the  pipe  of  a 


THE   BEGINNINGS   OF   EXPERIMENTATION       55 

pump  the  water  would  never  rise  over  32  feet.  The 
piston  of  an  ordinary  lift  pump  draws  the  air  out  of 
the  pipe  and  thus  creates  a  vacuum  into  which  the 
water  rushes,  as  was  early  recognized.  It  was  Hiero 
who  first  attributed  this  phenomenon  to  "  nature's 
abhorrence  of  a  vacuum, "  and  in  default  of  any  better 
explanation  this  was  accepted  until  Torricelli  investi- 
gated the  subject  experimentally. 

The  reasoning  which  led  Torricelli  to  his  famous 
experiment  is  as  follows:  If  air  has  weight,  then  on  the 
surface  of  the  water  in  a  well  there  is  the  downward 
pressure  of  the  air.  If  the  air  is  removed  from  a  part 
of  this  surface,  as  for  example  that  inclosed  by  the  pipe 
of  the  pump,  the  pressure  of  the  air  on  the  rest  of  the 
surface  of  the  water  should  force  water  up  the  pipe. 

This,  then,  was  Torricelli' s  theory  to  account  for  the 
action  of  a  suction  pump.  How  was  he  to  test  it? 
If  the  pressure  of  the  air  at  the  surface  of  the  earth  was 
the  cause,  and  not  some  mysterious  effect  of  a  vacuum, 
then  this  pressure  should  support  a  column  of  mercury 
only  one  thirteenth  of  32  feet,  since  mercury  is  thirteen 
times  as  dense.  If  this  deduction  corresponds  with 
observed  phenomena,  the  theory  has  been  corroborated 
although  not  necessarily  proven.  To  establish  it  a 
further  experiment  is  necessary. 

He,  therefore,  selected  a  tube  about  4  feet  long  and 
sealed  one  end.  This  he  filled  with  mercury,  closed 
the  open  end  by  his  finger,  and  inverted  the  tube, 
immersing  the  unsealed  end  in  a  basin  of  mercury. 
When  he  removed  his  finger  the  mercury  in  the  tube 
sank,  thus  forming  a  "  Torricelli  vacuum."  This 
vacuum  was  evidently  unable  to  hold  the  mercury,  which 


56  THE   REALITIES   OF   MODERN   SCIENCE 

fell  away  from  the  top  of  the  tube  until  its  column  was 
about  30  inches  high,  i.e.  about  one  thirteenth  of  32 
feet. 

The  theory  received  further  support  very  shortly 
from  an  experiment  by  Pascal.  He  reasoned  that  if 
the  barometer  was  carried  up  a  mountain  so  as  not  to 
be  so  deep  in  the  ocean  of  air  at  the  bottom  of  which 
we  live,  then  the  pressure  of  the  air  should  be  smaller 
and  it  should  be  unable  to  sustain  so  high  a  column  of 
mercury.  At  his  suggestion  his  brother-in-law  tried 
the  experiment  on  Puy  de  Dome,  and  wrote  back  to 
Pascal  that  he  was  "  ravished  with  admiration  and 
astonishment"  when  he  found  that  after  he  had  as- 
cended the  mountain  for  about  three  fifths  of  a  mile 
the  mercury  column  was  three  inches  shorter.  Of 
course,  if  an  air  pump  had  been  available  Torricelli's 
theory  might  have  been  checked  as  in  classrooms  to-day 
by  surrounding  both  the  tube  and  the  basin  of  mercury 
by  a  glass  vessel  and  then  evacuating  the  latter.  If 
the  experiment  is  performed  it  will  be  found  that  as 
the  air  is  removed  the  barometer  column  sinks  until 
it  is  level  with  the  surface  of  the  mercury  in  the  basin. 
As  air  is  allowed  to  reenter  the  vessel  the  column  rises 
to  its  original  height.1 

The  balance,  Torricelli's  barometer,  and  the  air 
pump,  which  followed  shortly,  are  three  of  the  scientific 
instruments  which  made  possible  much  of  the  experi- 
mental work  of  the  last  300  years.  The  microscope 
and  the  telescope,  the  latter  invented  by  Galileo,  have 

1  Boyle  performed  this  experiment  in  1659,  five  years  after  von 
Guericke  invented  the  air  pump.  Cf.  Moore,  "History  of  Chemis- 
try," McGraw-Hill  Book  Co.,  1918. 


THE   BEGINNINGS  OF   EXPERIMENTATION       57 

been  important  in  biology  and  astronomy.  The 
thermometer,  also  invented  by  Galileo,  and  the  manom- 
eter complete  the  list  of  the  most  important.  The 
last  is  an  instrument  for  measuring  the  pressure  of 
gases  and  is  a  development  of  the  apparatus  used  by 
Boyle  in  his  famous  demonstration  that  ah-  has  elas- 
ticity of  volume. 

Boyle's  experiments  on  the  "  spring  of  the  air," 
as  he  called  it,  may  be  illustrated  by  the  apparatus  of 
Fig.  4.  Some  air  is  trapped  in  the  short  and 
sealed  end  of  the  U-shaped  tube  by  pouring 
mercury  into  the  long  end.  If  the  heights  of 
the  mercury  in  the  two  arms  of  the  tube  are 
the  same,  the  air  in  the  closed  end  is  at  atmos- 
pheric pressure.  The  column  of  atmosphere 
acting  on  the  mercury  of  the  open  arm  is 
equivalent  to  the  column  of  mercury  of  a 
barometer.  For  convenience  in  describing  the  FlG 
experiment  let  us  assume  that  this  is  30  inches. 
If  more  mercury  is  poured  into  the  long  tube  the  levels 
are  altered.  The  pressure  on  the  trapped  air  is  greater 
than  atmospheric  pressure  by  the  amount  which  the 
column  of  mercury  in  the  open  tube  exceeds  that  in 
the  other.  If  this  distance  is  6  inches,  the  total  pres- 
sure on  the  inclosed  air  is  that  of  36  vertical  inches 
of  mercury.  The  volume  of  this  air  is,  of  course,  re- 
duced. The  new  volume  will  be  found  to  be  f£  of  the 
original  volume.  In  general,  if  the  pressure  is  changed 
the  volume  is  changed  inversely,  a  fact  known  as 
Boyle's  Law. 

We  now  see  that  such  a  device  may  be  used  as  a 
manometer  by  connecting  its  long  arm  to  the  container 


58 


THE  REALITIES  OF   MODERN  SCIENCE 


of  the  gas  for  which  we  wish  to  know  the  pressure. 
By  Boyle's  Law,  then,  we  may  calculate  the  pressure 
from  the  new  volume  of  the  air  in  the  short  arm. 

In  describing  this  experiment  on  Boyle's  Law  and 
its  application  to  the  construction  of  a  manometer  we 
have  tacitly  assumed  that  the  temperature  has  been 
kept  constant.  As  we  shall  see  later,  and  as  the 
reader  realizes,  changes  in  temperature  will  cause 
changes  in  the  volume  which  the  air  occupies,  apart 
from  any  change  in  pressure.  We  have  also  assumed 
that  the  tube  is  everywhere  the  same  size. 

If  a  tube  like  that  of  Fig.  5  is  used  the  same  experi- 
ment may  be  performed  with  identical  results.  Of 
course,  more  mercury  needs  to  be  added 
to  produce  the  same  difference  of  level. 
It  is  to  Pascal  that  we  owe  the  statement 
of  the  general  principle  which  underlies 
this  phenomenon.  To  understand  it  we 
distinguish  between  force  and  pressure, 
defining  pressure  1  as  the  force  per  unit 
area. 

Now  Pascal's  Law  states  that  fluids  at 
rest  transmit  pressure  equally  in  every 
Is  not  this  obviously  the  condition  which 
must  be  met  if  a  fluid  is  to  be  at  rest  ?  If  a  particle 
is  to  be  at  rest  it  must  sustain  pressures  from  the 
liquid  about  it  which  are  the  same  in  all  directions, 
for  if  more  pressure  is  transmitted  to  it  from  one  side 
than  the  other  it  will  move.  Conversely,  the  particle 

1  If  the  pressure  is  the  same  on  every  square  inch  of  cross-sec- 
tional area  then  the  total  force  exerted  is  the  product  of  the  area 
and  the  pressure. 


FIG.  5. 


direction. 


THE   BEGINNINGS   OF   EXPERIMENTATION       59 

itself  is  pushing  equally  in  every  direction,  for  if  it 
did  not  react  on  the  adjacent  particle  with  an  equal 
force  there  would  be  an  unbalance  of  force  which  would 
cause  motion.  In  other  words,  Pascal's  Law  means 
that  in  a  liquid  at  rest  any  two  adjacent  particles  of 
the  liquid  are  acting  on  each  other  with  equal  and 
opposite  forces.  Consider  now  two  adjacent  particles 
in  the  horizontal  portion  of  the  tube  of  Fig.  5.  Since 
the  liquid  is  at  rest  the  pressure  transmitted  from  the 
right  must  be  equal  to  that  from  the  left.  The  pres- 
sure exerted  by  the  fluid  in  the  long  tube  is  then  equal 
and  opposite  to  that  of  the  fluid  in  the  short  tube. 

The  total  downward  force  exerted  by  the  liquid  in 
the  right-hand  tube  is  as  many  times  greater  than  that 
of  the  left-hand  as  the  area  of  the  right  tube  is  greater 
than  the  area  of  the  left  tube.  A  mechanical  advan- 
tage may  therefore  be  obtained  by  utilizing  this  prop- 
erty of  a  liquid.  This  is  illustrated  by  the  hydraulic 
press  in  which  a  small  force  acting  on  a  piston  of  small 
area  produces  the  same  pressure  as  does  a  larger  force 
applied  to  a  correspondingly  larger  piston.  As  a  con- 
sequence of  the  incompressibility  of  liquids,  the  dis- 
tances through  which  the  acting  and  resisting  forces 
are  exerted  are  inversely  as  the  forces,  as  the  "work 
principle"  requires. 


CHAPTER  VI 

THE  REALITIES  OF  SCIENCE 

IN  the  classification  of  the  phenomena  which  are 
physical  rather  than  chemical,  it  was  natural  to  cor- 
relate them  according  to  the  senses  by  which  they  are 
perceived.  The  subjects  into  which  physics  has 
usually  been  divided  are  mechanics,  heat,  sound, 
light,  and  electricity  and  magnetism.  Thus  under 
"light"  and  " sound"  were  classed  those  natural 
phenomena  which  affected  the  optic  and  the  auditory 
nerves.  "Heat"  included  those  affecting  what  has 
been  called  the  "temperature  sense."  "Mechanics" 
included  phenomena  of  motion  and  of  forces.  Elec- 
tricity and  magnetism,  which  are  now  grouped  to- 
gether, were  for  a  long  time  considered  separate 
divisions.  They  dealt  with  the  motions  and  forces 
of  which  electricity  and  magnetism  are  the  causes. 

A  classification  according  to  senses  is  unsatisfactory. 
Light  does  not  exist  for  one  who  is  blind,  nor  sound  for 
one  who  is  totally  deaf.  Either  exists  only  in  so  far 
as  we  ourselves  are  concerned  and  have  certain  nerves. 
For  the  color-blind,  as  for  example  those  who  fail  to 
perceive  the  greens,  light  of  this  color  does  not  exist, 
although  it  may  for  others.  Those  of  normal  vision 
may  have  a  sensation  which  they  call  green  light,  but 
this  means  not  that  green  light  is  real  but  only  that  an 
impression  is  real  to  them.  Light  is  not  an  objective 
but  a  subjective  reality. 

60 


THE   REALITIES  OF  SCIENCE  61 

What  are  the  causes  of  the  sensations  which  we  call 
light?  What  is  the  objective  reality  of  such  phenom- 
ena? Let  us  attempt  to  answer  these  questions  by 
considering  in  detail  an  illustration  of  sound,  a  similar 
subjective  reality. 

We  are  all  familiar  with  some  musical  instrument  and 
know  that  sound  is  produced  from  a  stringed  instru- 
ment by  setting  the  string  in  motion.  Its  natural 
motion  is  periodic,  for  it  takes  the  same  time  for  each 
swing  back  and  forth  through  the  position  of  rest  from 
which  it  is  displaced.  The  smaller  the  period  of  this 
vibration,  the  higher  is  the  pitch  of  the  musical  note 
which  we  recognize. 

The  phenomena  are  most  easily  observed  it  produced 
in  a  slightly  different  manner.  Let  a  visiting  card  be 
held  so  as  to  touch  the  teeth  of  a  rotating  gear  wheel. 
The  card  is  pushed  away  and  flaps  back,  that  is,  vi- 
brates, once  for  every  tooth  which  comes  in  contact 
with  it.  If  the  wheel  is  turning  very  slowly  we  recognize 
the  flapping  of  the  card  as  a  regular  or  rhythmic  noise.  As 
the  speed  of  the  wheel  increases  this  becomes  a  musical 
note  of  rising  pitch.  The  number  of  complete  vibrations 
which  the  card  makes  in  each  second  is  called  its  "  vi- 
bration frequency"  and  is  the  measure  of  the  pitch. 

Consider  now  the  means  by  which  the  motion  of  the 
sounding  body  reaches  the  ear  of  the  listener.  The 
intervening  air  consists  of  small  discrete  particles  or 
molecules.  As  the  vibrating  card  is  pushed  out  by  a 
tooth  it  forces  ahead  of  it  the  adjacent  layer  of  mole- 
cules, which  in  turn  push  against  those  adjacent  to 
them.  A  city  crowd,  gathered  around  some  object 
of  interest,  as  it  surges  away  under  the  commands  and 


62  THE   REALITIES  OF   MODERN   SCIENCE 

shoves  of  the  policemen  at  the  center,  pictures  a  some- 
what similar  action.  The  push  or  pulse,  started  at  the 
center,  travels  outward  through  the  crowd.  Figure  6 

represents  a  series  of  layers. 
When  the  pulse,  which  origi- 
nates at  the  center,  reaches  a 
layer,  as  6  of  the  figure,  it 
results  in  a  momentary  crowd- 
ing of  this  layer  against  the 
next  outer  one.  The  layer  6  is 

thus  crowded  or  condensed  between  layer  5  which  is 
moving  outward  and  layer  7  which  has  not  yet  started  to 
move,  and  we  speak  of  this  pulse  as  one  of  condensation. 
Now  suppose  that,  just  after  those  at  the  center  have 
started  to  push  out,  conditions  change  so  that  they 
may  again  move  in.  They  do 
so,  increasing  the  distance  be- 
tween their  layer  and  the  next. 
This  layer  in  its  turn  moves 
inward,  and  successively  the 
other  layers  adjust  them- 
selves to  the  new  condition. 

Whereas  before  we  had  a  con-  '_ 

densation  we  now  have  a  pulse  of  rarefaction  spreading 
outward  from  the  center,  as  shown  in  Fig.  7  by  the 
relative  positions  of  the  layers  when  the  pulse  has 
reached  7.  In  the  figure  the  previous  pulse  of  con- 
densation is  shown  as  just  reaching  layer  13.  The 
vibrating  card  gives  rise  in  the  surrounding  air  to  a 
succession  of  alternate  condensations  and  rarefactions 
which  travel  outward  to  the  ear  of  the  listener.  The 
thin_drum  of  the  ear  is  pushed  in  by  each  pulse  of 


THE  REALITIES  OF  SCIENCE  63 

condensation  which  reaches  it  and  allowed  to  move  out 
by  each  succeeding  rarefaction. 

A  musical  note  is  seen  to  be  the  impression  of  a  listener 
whose  eardrum  is  moved  back  and  forth  periodically. 
Such  a  motion  may  be  obtained  from  a  vibrating  body 
through  the  medium *  of  the  intervening  air.  The  note 
depends  for  its  pitch  upon  the  frequency.  The  "  niter- 
national  pitch"  A  is  a  note  of  435  vibrations  per  sec- 
ond. When  the  frequency  is  above  about  20,000  per 
second  many  persons  are  incapable  of  hearing  it. 
Frequencies  of  30,000  or  more  are  inaudible  to  most 
persons.  These  high  notes  may  be  produced  by  small  or- 
gan pipes,  by  whistles,  by  vibrating  rods  which  are  very 
stiff  and  short,  and  by  other  means  similar  to  those 
used  to  produce  the  notes  of  music.  The  vibrations 
from  such  bodies  are  transmitted  to  the  ear  through 
the  air  hi  the  same  way  as  are  those  of  lower  frequency. 
In  one  case  we  say  there  is  the  sound  of  a  musical  note ; 
in  the  other  we  hear  nothing.  The  difference  is  in  our 
own  brains,  for  sound  is  a  subjective  reality. 

What  is  the  objective  reality  with  which  we  have  to 
deal,  which  is  present  hi  both  cases,  whether  we  hear 
the  sound  or  not  ?  The  most  obvious  reali ty  is  matter, 
both  in  the  case  of  the  vibrating  body  and  in  the  ah* 
which  transmits  these  vibrations.  Then  there  is  the 
motion  of  the  vibrating  body  and  of  the  surrounding 
ah-.  Shall  we  take  motion2  as  the  other  reality?  Be- 
fore we  decide  let  us  see  what  we  mean  by  motion. 

1  In  elementary  courses  on  physics  it  is  usual  to  demonstrate  the 
function  of  the  air  by  inclosing  the  source  of  sound  in  a  chamber 
which  is  then  evacuated. 

2  "Motion"  is  here  used  in  its  root  significance  and  with  no  con- 
notation of  "momentum." 


64  THE  REALITIES  OF  MODERN  SCIENCE 

In  speaking  of  motion  we  refer  the  position  of  one 
body  to  that  of  another.  Consider  the  case  of  two 
railroad  trains  at  rest  on  parallel  tracks.  We  know 
that  it  is  difficult  for  a  passenger  in  one  train  to 
tell  by  looking  at  the  other  whether  his  own  is  starting 
or  not.  He  may  see  the  other  apparently  sliding  by, 
but  he  cannot  decide,  on  visual  evidence  alone,  as  to 
his  own  motion  unless  he  can  see  the  ground.  Even 
if  he  is  told  that  he  is  actually  in  motion  he  cannot 
determine  the  direction.  Visually  he  can  decide  only 
as  to  whether  or  not  there  is  a  relative  motion  of  the 
two  adjacent  trains.  This  fact  may  be  illustrated  by 
placing  two  pencils  side  by  side,  making  a  mark  on 
each,  and  seeing  in  how  many  ways  one  or  both  may 
be  moved  so  as  to  produce  the  same  motion  of  one  mark 
relative  to  the  other. 

The  idea  of  motion  is  best  expressed  more  abstractly. 
Let  two  points  be  always  connected  by  a  straight  line ; 
if  this  line  changes  in  length  or  direction  there  is  a 
relative  motion 1  of  the  two  points.  It  is  always  with 
relative  motions  that  we  deal,  as  in  the  case  of  a  body 
falling  with  reference  to  the  earth.  If  we  knew  two 
fixed  and  intersecting  lines  in  the  universe  we  could 
refer  to  these  the  positions  and  motions  of  all  bodies. 
But  what  do  we  mean?  fixed  with  reference  to  what? 
We  reply,  "Why,  absolutely  fixed,"  and  use  "abso- 
lutely "  in  its  technical  sense.  It  is,  however,  preferable 
to  consider  all  motion  as  relative. 

Recognizing  that  motion  is  relative,  let  us  imagine 

1  An  interesting  exposition  of  the  historical  and  philosophical 
aspects  of  mechanics  is  that  of  Mach,  "The  Science  of  Mechanics," 
Open  Court  Publishing  Co.,  1907. 


THE   REALITIES  OF  SCIENCE  65 

that  a  listener  sways  his  head  from  side  to  side  just  in 
time  with  the  motion  of  the  layer  of  air  adjacent  to 
his  eardrum.  It  was  the  presence  of  this  air,  moving 
the  drum  periodically,  which  we  saw  to  be  the  cause 
of  his  auditory  impressions.  With  reference  to  the 
ground  the  motion  of  his  eardrum  will  be  made  just 
what  it  was  when  he  was  hearing  the  sound,  but  now 
there  will  be  no  relative  motion  of  the  eardrum  and 
the  skull.  A  relative  motion  of  these  is  required  to 
produce  an  effect  on  his  auditory  nerves,  so  that  he 
now  hears  no  sound  although  the  conditions  of  the 
source  and  of  the  intervening  air  are  just  what  they 
were  before. 

Another  illustration  which  shows  both  the  subjec- 
tivity of  sound  and  the  relativity  of  motion  is  a  matter 
of  common  observation.  The  pitch  of  an  automobile 
horn  or  of  a  locomotive  whistle  is  higher  if  the  machine 
is  approaching  than  when  it  is  stationary.  The  prin- 
ciple of  this  phenomenon,  which  occurs  in  light  as 
well  as  in  sound,  is  known  as  Doppler's.  Conversely, 
if  the  whistle  is  receding  from  the  listener  the  pitch 
is  lower.  The  vibration  frequency  of  the  horn  does 
not  depend  upon  which  direction  the  machine  is  going 
with  reference  to  the  listener.  Why  should  the  pitch 
be  different?  The  number  of  pulses  reaching  the  ear 
each  second  is  greater  when  there  is  a  relative  motion 
of  the  listener  toward  the  machine.  This  is  most 
evident  if  we  think  in  terms  of  a  train.  In  fact  we 
usually  speak  of  a  condensation  and  its  succeeding 
rarefaction  as  a  "wave"  and  of  the  succession  of  these 
as  a  "tram  of  waves."  Any  one  who  ever  watched  a 
long  freight  train  realizes  that  more  cars  per  minute 


66  THE   REALITIES   OF   MODERN  SCIENCE 

pass  him  if  he  walks  toward  the  point  from  which  the 
train  is  coming. 

In  trying  to  determine  what  are  the  objective  realities 
in  the  case  of  sound  we  have  so  far  recognized  that 
one  reality  is  matter.  The  other  apparently  involves 
motion,  but  because  motion  is  relative,  it  has  seemed 
preferable  to  look  further.  The  other  reality  in  sound 
we  shall  now  see  to  be  energy,  the  ability  to  do  work. 
In  order  to  show  this  we  shall  consider  in  more  detail 
the  vibratory  motion  of  a  sounding  string. 

The  string,  originally  at  rest,  is  set  in  vibration  by 
momentarily  deforming  it,  as  by  picking  it.  Elas- 
ticity is  called  into  play;  and  when  the  stress  is  re- 
moved the  string  flies  back  to  its  original  position. 
It  passes  through  this  position,  bowing  out  in  the 
opposite  direction.  Elastic  forces  are  now  operative 
just  as  they  were  in  the  case  of  the  original  deforma- 
tion. The  string,  therefore,  swings  back  through  its 
center  of  vibration  and  assumes  a  symmetrical  form 
on  the  other  side. 

In  producing  the  original  deformation,  work  is  done 
against  the  elastic  forces,  and  the  stretched  string 
acquires  an  ability  to  do  work.  This  principle  was 
early  applied  practically,  in  the  bow  of  the  ancients, 
where  the  energy  was  used  to  impart  motion  to  an 
arrow.  If  the  vibrating  string  is  a  source  of  sound, 
the  matter,  to  which  motion  is  imparted,  consists  of 
the  adjacent  molecules  of  air  and  also  of  the  molecules 
of  the  string  itself.  With  each  vibration  of  the  string 
some  of  the  original  energy  is  imparted  to  adjacent 
molecules  and  transmitted  by  successive  layers  of 
molecules  away  from  the  vibrating  string.  The 


THE  REALITIES  OF  SCIENCE  67 

spherical  layers  are  successively  larger  and  consist  of 
a  successively  greater  number  of  molecules.  Since 
a  larger  number  must  be  set  in  motion,  the  motion 
imparted  to  each  molecule  is  smaller  the  farther  it 
is  away  from  the  vibrating  source.  This  decrease 
means  a  decrease  in  the  possible  motion  of  the  ear- 
drum and  hence  in  the  intensity  of  the  sound  as  a 
listener  assumes  positions  farther  away  from  the  source. 
At  any  instant,  however,  the  energy  imparted  by  the 
previous  vibrations  of  the  string  is  associated  with  all 
the  molecules  between  the  string  and  that  most  dis- 
tant spherical  surface,  which  the  first  pulse  has  just 
reached. 

This  is  why  the  vibrations  gradually  decrease  and 
finally  cease.  The  work  done  in  producing  the  orig- 
inal deformation  is  gradually  passed  on  by  the  string 
to  the  ah-  molecules.  Since  the  string  in  each  swing 
must  push  ahead  of  it  the  air  molecules,  it  will  not 
swing  quite  as  far  as  if  there  were  no  work  to  be  done 
in  moving  the  air.  The  amplitude  of  the  vibration  then 
damps  down  just  as  in  the  case  of  the  swinging 
pendulum.  In  fact,  the  string  and  the  pendulum  are 
but  two  illustrations  of  a  type  known  as  simple  har- 
monic motion. 

We  passed  over  the  question  of  why  the  string,  when 
released  from  the  initial  deforming  stress,  should  fly 
through  its  center  of  rest.  As  an  elastic  body  it  should 
return  to  this  position,  for  each  molecule  of  the  string 
is  urged  toward  its  unstressed  position.  In  this  position, 
of  course,  the  restoring  force  of  elasticity  is  zero  and 
does  not  affect  the  motion  which  has  been  imparted 
to  the  string.  The  fact  that  the  latter  continues  in 


68  THE   REALITIES  OF   MODERN   SCIENCE 

motion  through  the  unstressed  position  is  a  character- 
istic of  bodies  which  we  describe  by  saying  that  they 
have  inertia. 

We  know  by  experience  that  it  takes  work  to  set  a 
body  into  motion,  to  make  a  body  which  is  already  in 
motion  move  in  a  different  direction,  or  to  stop  it. 
In  deforming  the  string  against  the  elastic  forces  we 
give  it  the  ability  to  do  work,  which  it  does  in  setting 
its  particles  into  motion  toward  their  original  positions. 
When,  however,  these  particles  arrive  at  this  position 
they  are  in  rapid  motion  and  thus  in  turn  have  the 
ability  to  do  work.  This  they  do  against  the  elastic 
force  of  the  string,  which  becomes  effective  the  moment 
the  latter  passes  through  its  unstressed  position.  A 
deformation  opposite  to  the  original  one  is  thus  brought 
about.  It  is  also  smaller,  for  the  particles  cannot  do 
quite  as  much  work  in  assuming  it,  since  some  of  the 
original  energy  has  been  transmitted  away  by  the  air 
and  some  dissipated  in  the  frictional  motions  of  the 
particles  of  the  string  itself.  For  this  reason  the 
vibration  is  damped. 

If  a  vibrating  source  is  surrounded  by  some  other 
medium,  as  for  example  by  water,  similar  phenomena 
would  occur,  although  the  wave  train  would  not  travel 
at  the  same  speed  and  the  source  would  damp  down 
more  quickly.  All  phenomena,  comprising  a  vibrating 
source  and  an  elastic  medium,  by  the  vibrations  of  the 
molecules  of  which  energy  is  transmitted  away  from 
the  source,  should  be  classified  as  sound,  although  only 
under  special  conditions  can  the  vibrations  be  detected 
by  a  human  ear. 

The  realities,  for  which  we  have  searched  in  con- 


THE   REALITIES  OF  SCIENCE  69 

sidering  the  phenomena  of  sound,  are  now  seen  to  be 
matter  and  energy.  With  these  science  is  concerned 
and  about  them  we  shall  group  our  remaining  dis- 
cussion. They  are  the  realities  of  mechanics  and  of 
light,  although  in  the  latter  case  the  medium  is 
not  molecular  and  the  vibrating  bodies  are  electrons. 
We  shall  find  them  the  same  for  the  other  subdi- 
visions of  physics  and  for  all  physical  science,  including 
chemistry. 


CHAPTER  VII 
THE  MOLECULAR  COMPOSITION  OF  MATTER 

THAT  the  realities  with  which  science  deals  are  matter 
and  energy  was  not  appreciated  until  comparatively 
recently.  In  1777  the  indestructibility  of  matter  was 
established  by  experiments  of  Lavoisier,  a  French 
chemist.  In  1843  experiments  of  Joule  established  the 
indestructibility  of  energy.  The  experiments  of  the 
former  will  be  considered  in  this  chapter,  and  those 
of  the  latter  in  a  later  chapter.  It  is  upon  the  basis 
of  the  indestructibility  of  these  entities,  energy  and 
matter,  that  we  are  entitled  to  consider  them  the 
realities  of  science. 

A  body  of  any  kind  of  matter  may  be  divided  into 
very  fine  particles,  as  we  all  know,  just  as  a  lump  of 
stone  may  be  crushed  under  water  until  its  particles 
float  away  undiscernible  to  the  human  eye.  Pulveriz- 
ing stone  gives  merely  small  particles  of  stone  and  does 
not  produce  any  change  except  of  size.  The  particles 
of  water  vapor  in  the  air  about  us  are  even  smaller  than 
anything  which  can  be  seen  with  the  best  microscope, 
but  still  they  are  unaltered  in  kind.  In  the  past  it 
has  sometimes  been  argued  that  there  was  no  limit 
to  such  divisibility  of  matter,  that  matter  in  fact  was 
" infinitely  divisible."  On  the  other  hand,  there  were 
philosophers  like  Democritus  (420  B.C.)  who  held  that 
it  is  granular  in  its  composition,  consisting  of  small 

70 


THE   MOLECULAR   COMPOSITION  OF   MATTER     71 

similar  parts  which  cannot  be  further  divided,  as  is 
indicated  by  the  name  "  atoms." 

The  present-day  concept  of  the  structure  of  matter 
was  not  firmly  established  until  about  1860.  Accord- 
ing to  the  ideas  accepted  since  that  date  matter  may 
be  subdivided  into  small  similar  parts,  called  molecules, 
which  are  the  limit  of  divisibility  for  that  kind  of 
matter.  To  subdivide  any  kind  of  matter  into  parts 
smaller  than  this  perfectly  definite  particle,  is  to  alter 
the  kind  of  matter. 

The  molecule  of  starch  is  one  of  the  largest  of  which 
we  know,  and  yet  it  is  so  small  that  the  most  powerful 
microscope  will  not  permit  us  to  observe  it  by  eye. 
We  do  not  know  what  its  shape  may  be,  but  if  we 
imagine  it  inclosed  in  a  sphere  we  do  know  that  the 
diameter  of  this  sphere  is  about  one  two-hundredth  part 
of  the  diameter  of  the  smallest  particle  which  we  can 
see  with  the  microscope.  The  molecule  of  sugar,  which 
is  much  larger  than  many  of  the  other  kinds  of  mole- 
cules, is  only  about  one  ninth  the  diameter  of  the  starch 
molecule.  The  diameter  of  the  molecular  spheres  for 
the  molecules  of  the  gases  which  compose  the  air  we 
breathe  is  different  for  each  kind,  but  is  about  2X10~8 
cm.1  How  these  dimensions  may  be  obtained  we  shall 
see  hi  Chapter  XX. 

Matter  of  most  kinds  may  be  divided  into  particles 
even  smaller  than  its  molecules,  but  whenever  such 
division  takes  place  the  kind  of  matter  is  changed. 

1  The  symbol  10-8  means  -=^,1.6.0.000,000,01.   Very  large  or  very 

small  numbers  are  conveniently  expressed  "in  powers  of  ten."  Since 
1  cm.  is  0.394  in.,  it  follows  that  the  molecular  diameter  is  about 
8xlO~9  inch  or  eight  thousandths  of  one  millionth  of  an  inch. 


72  THE   REALITIES  OF   MODERN  SCIENCE 

Thus  the  molecule  of  water  may  be  divided  into  three 
smaller  parts,  of  which  two  are  similar  particles  of 
hydrogen  and  one  is  a  particle  of  oxygen.  In  the  same 
way  common  salt  is  divisible,  each  molecule  giving 
one  particle  of  a  metal  known  as  sodium  and  one  of 
chlorine.  The  latter  we  know  as  a  green  poisonous  gas, 
but  in  this  form  each  molecule  of  it  consists  of  two  of 
the  smaller  particles  which  enter  into  the  molecule  of 
salt.  These  particles  into  which  the  molecule  is  di- 
visible or  of  which  the  molecule  is  composed  are  called 
atoms.  Of  the  atoms  there  are  about  ninety  different 
kinds.  These  different  kinds  of  matter,  which  can- 
not be  further  decomposed  into  other  kinds,  the  ele- 
mentary substances,  are  spoken  of  as  "the  elements. " 

This  concept  of  an  atomic  composition  was  fore- 
shadowed by  the  early  Greek  philosophers  but  was  not 
confirmed  until  about  1802,  when  it  was  set  forth  with 
experimental  evidence  by  John  Dalton.  During  the 
preceding  centuries  the  natural  desire  of  men  to  explain 
matter  had  found  its  outlet  in  theories  untested  by 
experiment.  Matter  had  been  classified  by  its  proper- 
ties and  its  apparent  similarities  rather  than  by  its 
experimentally  determined  composition.  The  four 
qualities  by  which  it  was  generally  compared  were 
those  of  fire,  earth,  water,  and  air.  Any  chemical 
change  was  regarded  as  due  to  the  proportion  of  these 
" elements "  or  " principles"  which  were  added  to  the 
original  substance.  But  as  to  the  concept  of  the  "  es- 
sence" to  which  these  principles  were  added,  opinion 
was  divided ;  some  of  these  ancient  and  medieval  phi- 
losophers held  that  it  was  a  material  substance  and 
others  that  it  was  "ethereal." 


THE   MOLECULAR  COMPOSITION  OF   MATTER     73 

There  is  some  reason  to  believe  that  the  thinkers  of 
the  school  which  considered  the  essence  material  and 
hence  sought  for  it  by  experiment  either  formed  or 
were  among  the  early  alchemists.  The  origin  of 
alchemy  and  even  the  derivation  of  the  name  itself 
are,  however,  matters  as  to  which  historians  are  in 
doubt.  By  the  fourth  century  of  this  era,  historical 
evidence  seems  to  show  that  the  attention  of  the 
alchemists  had  become  centered  upon  the  problem  of 
the  transformation  of  baser  metals  into  gold  and  silver. 
This  they  attempted  to  accomplish  by  adding  or  sub- 
tracting some  of  the  elementary  principles  mentioned 
above.  Then"  experiments  thus  led  to  the  discovery 
of  some  new  compounds  and  also  to  methods  for  separat- 
ing different  substances.  Alchemy  was  largely  in  the 
hands  of  the  Arabs  until  the  llth  century,  when  it 
became  European.  Although  it  was  experimental  it 
was  essentially  only  descriptive.  Its  theory  was  also 
incoherent  and  mystical. 

Throughout  the  centuries  during  which  chemistry 
was  a  mystical  art  rather  than  a  science,  progress  was, 
of  course,  made  in  the  processes  of  industrial  chemistry 
by  "cut  and  try"  methods.  The  metallurgical  opera- 
tions of  smelting  and  refining  advanced.  The  art 
of  healing  progressed  during  this  same  period  and  by 
the  15th  century  chemical  preparations  in  medicine 
had  become  of  considerable  popular  importance.  The 
attention  of  the  alchemists,  therefore,  turned  from 
their  unsuccessful  search  for  gold  to  a  search  for  new 
medicines.  But  medicine  was  also  a  science  hi  its 
infancy,  one  of  superstitions,  mysteries,  and  untried 
theories.  Thus  as  late  as  1800  Davy,  famous  to-day 


74  THE   REALITIES  OF   MODERN  SCIENCE 

for  his  miner's  safety  lamp,  demonstrated  by  breathing 
nitrous  oxide,  that  is,  "  laughing  gas,"  that  it  was  not 
the  " principle  of  contagion7'  which  the  medical  men  of 
his  day  considered  it. 

The  development  of  chemistry,  as  medicine,  extended 
only  from  the  15th  to  the  17th  century.  During  the 
17th  and  18th  centuries  the  influence  of  the  alchemistic 
theories  was  decreasing.  With  the  discoveries  of 
Black,  Priestley,  Cavendish,  Scheele  and  others  in  the 
second  half  of  the  18th  century  the  mystical  character 
disappeared  and  it  was  possible  for  Lavoisier  and  Dalton 
to  lay  the  foundation  of  modern  chemistry. 

The  revolt  against  the  doctrines  of  the  medieval  ages 
was  really  started  by  Robert  Boyle.  In  1662  he  pub- 
lished his  "  Skeptical  Chemist."  In  this  he  denied  the 
accepted  theory  and  stated  that  all  substances  were 
either  elements,  which  could  not  be  further  decomposed, 
or  else  compounds  of  two  or  more  such  elements.  Com- 
pounds, according  to  his  hypothesis,  were  formed  by 
the  coalescence  of  small  particles  of  the  elements 
concerned.  The  corpuscles  of  any  one  kind  he  con- 
sidered to  have  an  affinity  for  the  corpuscles  of  another 
kind,  and  this  affinity  was  the  cause  of  the  formation 
of  the  compound  corpuscles.  The  affinities  postulated 
by  Boyle  were  not,  however,  the  likes  and  dislikes  and 
other  mental  attributes  sometimes  used  by  his  contem- 
poraries or  predecessors  to  explain  chemical  reactions. 

For  centuries  the  theory  of  the  four  elements  had 
failed  to  account,  even  to  its  adherents,  for  the  forma- 
tion of  the  known  products  of  chemical  reaction.  Ad- 
ditional principles  had,  therefore,  been  accepted  as 
capable  of  modifying  the  nature  of  matter.  Mercury 


THE   MOLECULAR   COMPOSITION  OF   MATTER      75 

was  supposed  to  add  luster  and  to  make  the  substance 
volatile.  Salt  conferred  fixity  rather  than  volatility 
in  the  presence  of  fire.  Sulphur  added  combustibility. 
Later  the  " principle  of  combustibility"  was  assumed 
to  be  due  to  other  causes  than  the  addition  of  sulphur, 
so  that  a  more  inclusive  term  came  to  be  desirable  and 
hi  1690  the  principle  was  named  " phlogiston." 

According  to  the  phlogiston  theory,  combustion 
meant  the  liberation  of  phlogiston,  and  those  substances 
which  burned  most  readily  released  the  larger  quan- 
tities. Charcoal,  wood,  and  coal  were  thus  assumed 
to  be  nearly  pure  phlogiston.  Now,  metals  exposed  to 
fire  tarnish  and  in  time  form  earthy  powders.  These 
we  call  oxides.  The  familiar  red  rust  of  iron  is  an 
oxide  formed  by  a  reaction  with  water  and  air.  Some 
of  the  iron  ores  are  oxides.  The  method  of  reducing 
metallic  oxides  by  heating  them  with  charcoal  had 
been  used  for  centuries.  The  reaction  which  takes 
place  is  one  of  combustion  of  the  charcoal  and  the 
oxygen  of  the  metallic  oxide.  Carbon  oxides,  the 
usual  products  of  combustion,  with  which  we  are  all 
familiar,  are  thus  formed  and  a  residue  of  pure  metal 
and  charcoal  ash  is  left.  The  advocates  of  the  phlogis- 
ton theory  curiously  hi  verted  this  idea  of  combustion. 
They  assumed  that  phlogiston  left  the  charcoal  and 
entered  the  oxide,  the  earthy  powder  which  they  called 
"calx."  The  calx  thus  became  a  metal  by  the  addition 
of  phlogiston.  Conversely,  when  a  metal  was  burned 
it  was  dephlogisticated  and  a  calx  resulted. 

The  opponents  of  the  phlogiston  theory  pointed  out 
that  the  calx  was  heavier  before  it  became  a  metal  by 
the  addition  of  phlogiston.  Any  such  objection  was, 


76  THE   REALITIES  OF   MODERN   SCIENCE 

of  course,  easily  to  be  met,  if  one's  science  was  a  matter 
of  qualities  and  principles,  by  describing  phlogiston 
as  a  " principle  of  levity"  such  that  it  decreased  the 
weight  of  the  body  into  which  it  entered.  In  the 
choice  of  such  additional  principles  the  advocates  of 
this  theory  were  not  always  consistent. 

The  problem  of  combustion  was  not,  however,  to  be 
solved  until  it  was  shown  that  air  was  a  mixture  of 
dissimilar  gases.  That  gases  did  differ  in  kind  and  in 
density,  that  is,  mass  per  unit  volume,  was  definitely 
established  by  Black's  experiments  in  1752  on  carbonic 
acid  (carbon  dioxide  and  water).  Black,  who  was  a 
Scot  and  a  student  of  medicine,  was  interested  early 
in  his  life  in  the  medicinal  properties  of  mineral  waters. 
In  his  study  of  limewater  he  found  that  limestone 
(now  known  as  a  compound  of  calcium,  carbon,  and 
oxygen,  CaC03)  when  heated  lost  a  gas  (carbon  dioxide, 
C02).  The  weight  of  the  gas  thus  produced  he  found 
by  the  balance  to  be  equal  to  the  loss  in  weight  of  the 
stone.  He  also  brought  about  the  reverse  reaction, 
obtaining  calcium  carbonate  from  lime  (calcium  oxide) 
and  carbonic  acid,  and  demonstrated  for  this  reaction 
the  conservation  of  mass. 

In  1773  Scheele,  a  Swedish  chemist,  discovered 
oxygen  as  a  constituent  of  air.  The  succeeding  year 
Priestley  also  discovered  oxygen.  The  name,  however, 
was  suggested  later  by  Lavoisier.  Priestley  obtained 
oxygen  by  heating  a  tube  of  red  oxide  of  mercury  by 
focusing  the  sun's  rays  upon  it  with  a  burning-glass. 
The  gas  thus  liberated  was  found  to  support  com- 
bustion better  than  did  ordinary  air.  For  example, 
a  glowing  ember  thrust  into  a  vessel  containing  oxygen 


THE   MOLECULAR   COMPOSITION  OF   MATTER^   77 

will  throw  out  sparks  and  burst  into  flame.  Priestley 
was  also  the  first  to  notice  the  exhilarating  effect  of 
breathing  pure  oxygen,  although  he  preceded  the 
experiment  upon  himself  by  trying  the  gas  on  two 
mice. 

The  experiments  of  Lavoisier  extended  over  several 
years.  His  final  conclusion  as  to  air  was  reached  in 
1777.  It  consists  of  a  mixture  of  two  gases,  one 
capable  of  supporting  combustion  and  the  other 
incapable,  oxygen  and  nitrogen,  respectively.  By 
this  time  he  had  coordinated  and  extended  the  re- 
searches of  his  contemporaries,  reaching  a  conclusion 
in  accord  with  their  experimental  results  but  not  always 
with  their  theories,  for  many  of  them  were  phlogis- 
tists.  His  demonstration  of  the  fallacy  of  the  phlogis- 
ton theory  occurred  earlier  (1772).  He  burned 
phosphorus  in  a  closed  vessel  and  showed  that  when 
all  the  phosphorus  which  could  be  made  to  burn  had 
been  consumed,  (1)  about  one  fifth  of  the  air  had 
disappeared  (the  oxygen) ;  (2)  the  loss  in  weight  of 
the  air  was  practically  equal  to  the  difference  in  weight 
of  the  resultant  white  solid  and  the  original  phosphorus ; 
and  (3)  the  density  of  the  residual  ah*  was  now  less 
than  that  of  the  original  ah-.  An  effective  demonstra- 
tion of  the  conservation  of  matter  was  thus  made  by 
Lavoisier.  The  progress  of  the  •"  science  of  the  com- 
position of  substances,"  as  Boyle  had  defined  chemistry, 
thereafter  proceeded  with  great  rapidity. 

The  discovery  of  the  chemical  composition  of  the 
atmosphere  as  one  volume  of  oxygen  to  each  four 
volumes  of  atmospheric  nitrogen1  gave  no  immediate 
1  Nitrogen  and  traces  of  some  other  gases. 


78  THE   REALITIES  OF   MODERN  SCIENCE 

clew  to  its  physical  structure.  The  union  of  oxygen 
and  nitrogen  was  supposed  by  many  scientists  to  be 
chemical,  one  substance  dissolving  the  other.  When 
water  evaporated  into  the  air  it,  also,  was  supposed  to 
be  dissolved  by  the  existing  compound  of  nitrogen  and 
oxygen.  This  idea  was  incomprehensible  to  Dalton. 
He  tried  to  visualize  such  a  mixture  in  terms  of  small 
particles  which  he  called  atoms,  adopting  the  ideas 
of  a  granular  structure.  The  difficulty  in  his  mind 
was  that  the  compound  particles  would  be  of  different 
masses  and  should  ultimately  settle  toward  the  surface 
of  the  earth,  resulting  in  a  stratified  atmosphere  just 
as  two  liquids  of  different  density  will  separate  under 
gravity.  It  had  been  found,  however,  by  testing  air, 
obtained  at  various  altitudes,  that  the  proportions  of 
oxygen  and  nitrogen  were  essentially  constant  and  hence 
that  no  such  phenomenon  of  stratification  occurred. 
Such  a  chemical  theory  could  not  be  reconciled  with 
the  observed  facts,  and  yet  Dalton  felt  the  need  of 
picturing  the  phenomenon  in  terms  of  corpuscles.  At 
first  it  seemed  to  him  equally  inconceivable  that  the 
air  should  be  merely  a  mechanical  mixture  of  particles 
of  the  two  gases  just  as  we  might  make  a  mixture  of 
baseballs  and  tennis  balls. 

The  pressure  of  a  contained  gas  had  been  shown  by 
Boyle  to  be  due  to  its  " spring'7  or  elasticity.  Ber- 
nouilli  in  1738  had  suggested  that  the  cause  of  the 
pressure  was  the  impacts  upon  the  surface  of  the  con- 
tainer of  the  small  particles  of  the  air.  The  kinetic 
theory,  which  explains  such  pressure  as  due  to  molecu- 
lar impacts  and  the  motions  of  the  molecules  as  due 
to  the  energy  which  they  possess,  was  not,  however, 


THE   MOLECULAR   COMPOSITION  OF   MATTER     79 

advanced  in  an  acceptable  form  until  the  work  of 
Clausius  in  1857.  In  Dalton's  day  the  effect  which 
Boyle  had  noticed  was  explained  on  the  basis  of  forces. 
This  was  largely,  of  course,  because  of  the  influence 
of  Newton's  concept  of  forces.1  For  the  present  it  is 
only  necessary  to  note  that  the  pressure  was  supposed 
to  be  due  to  the  interactions  of  the  various  corpuscles. 
These  were  said  to  repel  each  other  with  a  force  which 
became  smaller  as  the  separation  between  the  particles 
increased.  If  both  kinds  of  particles  repelled  then 
stratification  was  still  possible.  But  if  the  only  actions 
were  between  similar  particles  then  each  gas  would 
tend  to  expand  and  fill  any  available  space.  Dal  ton, 
therefore,  assumed  that  each  gas  acted  as  a  vacuum 
so  far  as  the  other  gas  was  concerned. 

This,  as  we  shall  see,  is  also  the  modern  view,  but  we 
recognize  to-day  that  it  is  only  approximately  true  and 
depends  upon  the  actual  amounts  of  the  gases  as  com- 
pared to  the  volume  of  the  container.  If,  compared  to 
this  latter  volume,  the  space  occupied  by  the  gas  mole- 
cules is  small,  then  their  motions  are  practically  in- 
dependent of  each  other's  presence.  The  phenomenon 
has  a  parallel  in  the  case  of  a  dancing-floor.  If  the 
space  occupied  by  the  couples  while  at  rest  is  small  as 
compared  to  the  floor  area,  then*  motions  hi  dancing 
will  be  uninfluenced  by  each  other's  presence. 

For  pressures  not  greater  than  a  few  atmospheres 
Dalton's  assumption  would  be  true.  He  verified  it 
experimentally  and  announced  his  law  of  partial 
pressures,  that  the  pressure  exerted  by  a  contained 

1  The  laws  of  Newton  and  the  modern  theory  as  to  gas  pressures 
will  be  considered  in  Chapters  XII  and  XIII,  respectively. 


80  THE  REALITIES  OF   MODERN  SCIENCE 

volume  of  air  or  other  mixture  is  the  sum  of  the  pressures 
which  each  component  gas  would  exert  if  contained 
alone  in  a  similar  volume.  Dalton  was,  therefore, 
able  to  visualize  a  mixture  of  gases  as  a  purely  mechan- 
ical mixture  of  small  particles,  all  those  of  each  con- 
stituent gas  being  alike. 

He  called  all  such  particles  " atoms"  and  failed  to 
make  the  modern  distinction  between  the  molecular 
and  atomic  particles.  Such  a  distinction  was  first 
pointed  out  by  Avogadro  in  1811.  The  classification 
of  Avogadro,  however,  was  not  generally  accepted 
during  the  next  few  years,  and  considerable  confusion 
resulted.  In  fact,  the  confusion  as  to  the  concepts  of 
atom,  molecule,  and  atomic  weight  gave  rise  to  such 
conflicting  theories  and  methods  of  expressing  chemical 
composition  that  the  development  of  the  science  was 
impeded.  A  conference  was  therefore  called  at  Karls- 
ruhe in  1860  to  discuss  the  various  hypotheses.  A 
paper  containing  the  researches  and  theories  of  Canniz- 
zaro,  a  professor  at  Rome,  which  had  been  published 
in  1858,  was  brought  to  attention.  With  the  acceptance 
of  his  conclusions  modern  chemistry  became  established. 

If  we  accept  the  idea  that  the  molecules  of  any  kind 
of  matter  are  similar  and  are  formed  by  the  combina- 
tion of  atoms  of  different  elements,  all  the  atoms  of 
any  element  being  alike,  we  reach  at  once  certain 
conclusions.  Thus,  suppose  that  we  ask  what  possible 
compounds  may  be  formed  of  two  elements,  say  A  and 
B.  The  simplest  molecule  would  be  formed  by  one 
atom  of  A  and  one  of  B.  Such  a  compound  molecule 
we  might  then  symbolize  as  AB.  The  molecule  might 
be  formed  of  one  atom  of  A  and  two  of  B,  in  which  case 


THE   MOLECULAR   COMPOSITION  OF   MATTER      81 

it  would  be  represented  by  AB2,  where  the  subscript 
represents  the  number  of  atoms,  of  the  kind  indicated 
by  the  letter,  which  enter  into  the  formation  of  the 
molecule.  Other  possible  combinations  are  A2B,  A2B2, 
A2B3,  AsB,  AsB2,  and  so  on.  Whether  or  not  any 
particular  one  of  the  compound  molecules  which  we 
have  represented  can  be  formed  will  depend  upon  the 
mechanism  by  which  the  atoms  of  a  molecule  are  held 
together.  This  we  shall  discuss  in  the  next  chapter. 

Let  us  now  see  what  experiment  Dalton  performed  to 
test  such  an  assumption  as  to  molecular  composition. 
In  1802  he  discovered  that  if  he  mixed  100  parts  of 
common  air  with  36  parts  of  nitric  oxide  he  could  obtain 
a  combination  of  all  the  oxygen  of  the  air  with  this 
oxide,  leaving  a  residue  of  79  parts  of  atmospheric 
nitrogen.  Such  a  combination l  occurred  if  the  mixture 
was  made  in  a  narrow  vessel  or  tube.  On  the  other 
hand,  if  the  combination  took  place  hi  a  large  vessel 
over  water,  in  which  case  it  would  be  very  rapid,  a 
different  compound2  was  obtained.  In  this  case  he 
could  mix  100  parts  of  air  with  72  parts  of  nitric  oxide 
and  all  the  oxygen  would  enter  into  combination,  as 
would  be  evidenced  by  the  reduction  of  the  air  to  79 
parts  as  before.  Evidently  the  same  amount  of  oxygen 
could  enter  into  combination  with  a  definite  amount 
of  nitric  oxide  or  with  twice  that  amount.  But  Dalton 

1  Nitric  oxide  is  NO.     This  compound  is  nitric  peroxide,  or  in 
symbols  NO2.     Thus  2NO+O2=2NO2. 

2  The  compound  thus  formed  was  nitrous  anhydride  or  N2O3. 
Four  molecules  of  NO  combine  with  one  of  oxygen,  containing  two 
atoms,  as  may  be   expressed  in  the  equation  4NO+O2=2N2OS. 
Other  compounds  which  may  be  formed  are  NaO  or  nitrous  oxide, 
and  N2O6  or  nitric  anhydride. 

Q 


82  THE   REALITIES  OF   MODERN  SCIENCE 

showed  that  it  had  to  be  either  one  or  the  other.  Thus 
if  he  supplied  more  than  72  parts  of  nitric  oxide  to  100 
parts  of  air  there  was  a  corresponding  residue  of  nitric 
oxide  as  well  as  of  nitrogen.  Similarly  if  less  than  72 
parts  were  employed  the  residue  contained  oxygen, 
indicating  that  the  amount  of  nitric  oxide  available 
was  insufficient  to  form  compound  particles  with  all 
the  particles  of  the  21  parts  of  oxygen. 

By  this  and  other  similar  experiments  Dalton  estab- 
lished the  granular  composition  of  matter.  He  sum- 
marized these  phenomena  in  his  law  of  "multiple 
proportions,"  the  value  of  which  to-day  is  largely 
historical.  It  was  the  first  satisfactory  evidence  as 
to  the  molecular  and  atomic  structure  of  matter.  If 
we  start  our  study  by  accepting  the  concept  of  atoms 
and  molecules  we  need  not  burden  our  minds  with  its 
formal  expression.  Combinations  of  atoms  into  mole- 
cules must  always  involve  whole  numbers  of  atoms. 
Molecules  which  differ  in  the  number  of  similar  atoms 
which  they  contain  must  differ  in  properties  both 
physical  and  chemical.  The  converse  is,  however, 
not  always  true.  It  may  happen,  particularly  in  the 
case  of  molecules  involving  many  similar  atoms,  that 
different  substances  may  be  formed  by  the  same  com- 
bination of  atoms,  just  as  different  words  may  be 
formed  by  the  same  combination  of  letters.  The  same 
combination  of  atoms  as  forms  the  molecule  of  alcohol 
may  also  be  formed  into  a  molecule  of  " methyl  ether," 
an  ether  somewhat  like  the  anaesthetic. 

After  the  molecular  composition  of  matter  had  been 
demonstrated  by  Dalton  it  was  natural  to  assume  that 
equal  volumes  of  dissimilar  gases  would  contain,  under 


THE   MOLECULAR   COMPOSITION  OF   MATTER     83 

similar  conditions  of  pressure  and  temperature,  equal 
numbers  of  molecules.  As  a  matter  of  fact,  Gay- 
Lussac  found  that  gases  combined  in  simple  proportions 
by  volume.  He  was  unable,  however,  to  reach  the 
conclusion  that  the  number  of  molecules  hi  equal 
volumes  were  equal,  because  of  apparently  conflicting 
experimental  evidence.  For  example,  if  equal  volumes 
of  hydrogen  and  chlorine  gas  are  combined  the  result- 
ing volume  under  the  same  conditions  of  pressure  and 
temperature  is  found  to  be  the  sum  of  the  original 
volumes.  If  each  hydrogen  molecule  combined  with  a 
chlorine  molecule  there  should  be  only  as  many  mole- 
cules as  there  were  originally  of  either  chlorine  or 
hydrogen,  and  hence  we  should  expect  the  combination 
to  occupy  only  the  volume  originally  occupied  by  either. 
Avogadro,  however,  assumed  that  the  number  of 
molecules  must  be  proportional  to  the  volume  and, 
since  the  volume  of  the  compound  was  twice  that 
of  each  constituent,  that  there  were  twice  as  many 
molecules  of  hydrochloric  acid.  But,  since  a  mole- 
cule of  hydrochloric  acid  must  contain  a  particle  of  hy- 
drogen and  a  particle  of  chlorine,  there  must  have 
been  originally  twice  as  many  particles  of  both  hydrogen 
and  chlorine  as  there  appeared  to  be  from  the  volume 
they  occupied.  If  each  particle  of  hydrogen  really 
consisted  of  two  smaller  particles,  and  similarly  for 
chlorine,  then  the  experimental  facts  of  the  formation 
of  hydrochloric  acid 1  would  not  be  contradictory  to  his 

1  In  chemical  notation  the  formation  of  hydrochloric  acid  is  ex- 
pressed as  H8+Clj=2HCl,  indicating  that  each  molecule  of  hydrogen 
and  of  chlorine  is  diatomic  and  that  the  combination  of  a  molecule 
of  each  kind  results  in  two  molecules  of  hydrochloric  acid. 


84  THE   REALITIES  OF   MODERN  SCIENCE 

assumption.  He,  therefore,  announced  that  equal 
volumes  of  different  gases  at  equal  pressures  and  tem- 
peratures contained  equal  numbers  of  molecules.  He 
also  postulated  the  modern  distinction  between  atoms 
and  molecules  and  stated  that  the  molecules  of  an 
element  might  be  compounds  of  the  atoms  of  the 
element. 

Avogadro's  assumption  as  to  numbers  of  molecules 
offered  a  very  convenient  method  for  comparing  the 
masses  of  the  molecules  or  atoms  of  different  substances 
provided  they  were  in  the  gaseous  form.1  It  is  possible 
to  make  comparisons  only,  so  the  mass  of  the  molecule 
or  atom  of  some  gas  must  be  taken  as  a  standard. 
Hydrogen  as  the  lightest  gas  was  so  chosen.  Con- 
fusion, however,  resulted  until  Cannizzaro's  work, 
because  some  investigators  compared  with  the  atom 
and  others  with  the  molecule.  Berzelius,  a  chemist 
of  influence,  who  investigated  relative  atomic  weights, 
denied  Avogadro's  idea,  which  conflicted  with  his 
own  assumptions  of  an  electrical  attraction  between 
the  particles  which  combine  in  a  chemical  reaction. 
His  theories  along  this  line  have  since  proved  to  be  erro- 
neous and  need  not  be  discussed. 

The  atomic  weight  of  hydrogen  was  chosen  as  the 
standard  and  called  unity.  Upon  this  basis  the  molec- 
ular weight  of  hydrogen  gas  is  2.  Oxygen  was  found 
to  be  approximately  16  times  heavier  than  hydrogen, 
so  that  its  molecular  and  atomic  weights  are  about 
32  and  16,  respectively.  Later  it  was  agreed  to  express 

1  The  atomic  weights  of  non-gaseous  elements  are  obtained  by 
weighing  the  constituents  and  the  products  of  various  chemical 
reactions. 


THE   MOLECULAR   COMPOSITION  OF   MATTER     85 

all  atomic  weights  in  terms  of  oxygen  taken  as  16  rather 
than  hydrogen  as  1.  The  atomic '  weight  of  hydrogen 
as  expressed  on  this  scale  is  1.008. 

If  we  wish  to  deal  always  with  the  same  number  of 
molecules  we  may  do  so,  then,  by  taking  of  each  sub- 
stance a  number  of  grams  equal  to  its  molecular  weight. 
Thus  32  grams  of  oxygen  and  2.016  grams  of  hydrogen 
contain  the  same  number  of  molecules.  Such  an 
amount  of  any  substance  is  known  as  "1  gram-mole- 
cule/' or  by  abbreviation  as  "1  gm.-mole,"  or  more 
simply  as  "1  mole." 

The  combination  of  atoms  in  groups,  which  we 
symbolized  in  terms  of  A  and  B  earlier  in  this  chapter, 
represent  of  course  but  a  few  of  the  combinations 
which  we  can  imagine.  Why  do  some  of  these  com- 
binations occur  hi  chemical  reactions  and  not  others? 
Why,  for  example,  does  one  hydrogen  atom  combine 
with  one  chlorine  atom  to  form  hydrochloric  acid,  HC1, 
but  two  hydrogen  atoms  combine  with  one  of  oxygen 
to  form  water,  H2O?  Why  do  we  not  obtain  a  com- 
pound like  HC12  or  H2C1?  What  is  the  cause  of  such 
combinations  ? 

i  Speculations  as  to  the  cause  have  been  made  since 
the  time  of  the  early  Greek  philosophers.  Hippoc- 
rates assumed  that  "like  draws  to  like"  and  that 
substances  which  combined  had  something  in  common. 
The  early  Greek  atomists  personified  the  atoms,  at- 
tributing to  them  loves  and  hates.  The  medieval 
alchemists  were  more  picturesque  in  their  ideas.  They 
assumed  that  the  combinations  were  due  to  differences 
in  the  forms  of  the  particles  of  substances.  They 
pictured  acids  as  composed  of  sharp  particles  like 


86  THE   REALITIES  OF   MODERN  SCIENCE 

needles  or  spears,  which  transfixed  the  particles  of 
the  substances  upon  which  the  acid  acted.  Newton 
explained  the  actions  in  terms  of  forces  of  attraction 
which  were  negligible  except  when  the  atoms  between 
which  they  acted  were  very  close  to  each  other.  With 
the  discovery  of  Volta1  in  1800  of  a  relation  between 
chemistry  and  electricity,  theories  like  that  of  Berzelius 
arose.  Although,  as  we  shall  see  in  the  next  chapter, 
present  theories  point  to  an  electrical  cause,  the  details 
of  the  mechanism  are  largely  unknown. 

It  became  convenient  to  speak  of  the  combining 
ability  of  atoms  without  particular  reference  to  any 
theory  of  affinity.  The  term  "  valence  "  thus  came  into 
use.  One  atom  of  chlorine  combines  with  one  of 
hydrogen,  and  no  combinations  have  been  formed, 
the  molecules  of  which  contain  only  hydrogen  and 
chlorine  atoms,  in  which  more  than  one  atom  of  each 
element  was  involved.  These  two  substances  are 
said  to  have  similar  power  of  combination  or  valence. 
Oxygen  has  twice  the  valence  of  hydrogen  or  chlorine, 
for  one  atom  of  oxygen  combines  with  two  of  hydrogen. 

Valence  is  always  referred  to  that  of  hydrogen,  which 
is  taken  as  unity.  Chlorine  is  thus  seen  to  be  mono- 
valent  while  oxygen  is  divalent.  A  monovalent  atom 
may  for  the  moment  be  likened  to  a  one-armed  man 
who  may  grasp  the  hand  of  another  one-armed  man, 
as  in  HC1.  An  oxygen  atom  in  such  a  picture  has  two 
hands  and  may  grasp  those  of  two  monovalent  atoms 
such  as  hydrogen.  Carbon  has  four  bonds.  When  it 
burns  incompletely  we  have  the  formation  of  the 
extremely  poisonous  carbon  monoxide,  CO.  Two  of 

1  Cf .  Chapter  XIV. 


THE   MOLECULAR   COMPOSITION  OF   MATTER     87 

the  bonds  of  the  carbon  are  satisfied  by  the  two  of  the 
oxygen  atom,  but  the  remaining  two  are  available  for 
other  combinations.  This  is  apparently  the  cause  of 
its  effects  when  inhaled.  On  the  other  hand,  if  the 
combustion  of  the  carbon  is  complete  all  four  bonds 
are  satisfied  by  two  atoms  of  oxygen  and  the  resulting 
compound  is  carbon  dioxide, 


CHAPTER  VIII 
THE  ELECTRON 

THE  electron,  discovered  in  the  last  years  of  the  19th 
century,  has  indicated  the  electrical  composition  of 
matter.  Although  we  spoke  of  the  realities  of  science 
as  matter  and  energy,  we  might  equally  well  speak  of 
electricity  and  energy,  as  we  shall  see  in  succeeding 
chapters.  The  concept  of  the  electron  has  already 
explained  many  perplexing  phenomena  and,  of  course, 
raised  new  questions  itself.  The  atoms  of  chemistry 
are  no  longer  regarded  as  indivisible  particles.  An 
atom,  we  have  reason  to  believe,  always  consists  of  a 
number  of  electrons  and  another  part  which  is  called 
the  "  nucleus. "  These  electrons  are  little  bits  of 
electricity.1 

Further  than  to  say  that  electrons  are  electricity 
we  cannot  go.  We  can  say  that  matter  is  molecular, 
that  molecules  are  composed  of  atoms,  and  that  atoms 
are  formed  of  electrons.  In  finding  how  the  matter 
of  the  universe  is  composed  scientists  have  at  last 
reached  the  electron.  In  terms  of  it  they  can  explain 

1  But  what  is  electricity  ?  Sometimes  it  is  supplied  to  us  over 
wires  from  a  power  plant  and  utilized  in  a  wire  device  like  an  elec- 
tric lamp.  What  then  happens  is,  that  the  power  plant  pumps,  or 
forces,  a  procession  of  these  electrons  through  the  wires,  making 
them  hot  and  that  of  the  lamp  white  hot.  The  heat  comes  from 
the  work  done  by  the  stream  of  electrons  in  their  passage  through  the 
wire. 

88 


THE  ELECTRON  89 

fairly  well  everything  else,  or  at  least  there  is  promise 
that  ultimately  everything  else  will  be  so  explained. 
But  as  to  the  electron  itself  no  explanation  can  be 
given.  If  any  explanation  is  ever  obtained  it  will  be 
in  terms  of  something  else  which  in  its  turn  will  be  un- 
explainable  and  have  to  be  accepted  as  the  fundamental 
element  or  beginning  from  which  all  other  explanations 
start.  The  electron  is  the  fundamental  entity  with 
which  modern  science  starts. 

But  what  is  the  nucleus?  That  we  do  not  as  yet 
know.  To  learn  what  we  do  know  of  it  we  must  con- 
sider further  the  behavior  of  electrons.  These  are,  so 
far  as  we  know,  all  alike  without  regard  to  the  atoms 
from  which  they  are  derived.  An  electron  from  a 
hydrogen  atom  is  just  the  same  as  one  from  an  atom 
of  copper  or  one  from  an  atom  of  radium.  They  are 
very  small1  as  compared  to  an  atom  although  they 
are  for  their  size  much  heavier.  When  it  comes  to 
moving  an  electron,  as  for  example  starting  one,  it  is 
found  that  it  has  about  T^  of  the  inertia  of  the  hy- 
drogen atom.  If  you  realize  that  it  takes  about 
1.70X1025  hydrogen  atoms  to  weigh  an  ounce  you  will 
see  how  small  is  the  mass  of  an  electron. 

Now,  electrons  have  a  peculiar  property  of  repelling 
each  other.  Two  electrons  always  exert  a  force  push- 
ing each  other  apart  even  though  they  may  not  be  in 
contact.  How  can  an  electron  which  is  not  touching 
another,  push  on  it,  urging  the  second  electron  away 
from  itself?  Again  the  answer  is  that  we  do  not  know. 
They  must  act  on  each  other  through  a  "medium,"  a 
"something  between."  Such  a  medium  the  physicist 
1  What  we  mean  by  "size"  is  discussed  a  little  later. 


90  THE   REALITIES  OF  MODERN   SCIENCE 

must  assume,  for  of  the  repulsion  he  has  positive 
proof.  He,  therefore,  postulates  a  medium  which  he 
calls  the  "aether,"  or,  as  now  spelled,  "ether."  The 
word,  derived  from  the  Greek  "belonging  to  the 
upper  air,"  was  originally  applied  to  the  medium  be- 
tween the  earth  and  the  heavenly  bodies.  The  ether 
is  imponderable  and  intangible  and  we  cannot  detect 
its  existence  by  our  senses. 

When  the  physicist  speaks  of  a  vacuum  he  means  a 
space  free  from  matter,  but  through  it  electrical  repul- 
sions are  exerted  and  light,  heat,  and  radio-telegraph 
waves  may  pass.  Through  such  ethereal  spaces  the 
light  of  the  sun  and  the  stars  reaches  our  earth.  To 
the  modern  physicist,  these  vacua  are  empty  only  as 
far  as  concerns  matter,  that  is,  molecules,  atoms,  and 
dislodged  electrons.  According  to  his  ideas  a  vacuum 
is  full  of  ether.  In  fact,  he  considers  all  space  through- 
out the  universe  to  be  filled  continuously  by  this  ether, 
like  an  enormous  ocean,  in  which  exist  as  specks  the 
electrons  and  the  atoms  which  they  form.  That  the 
electrons  are  really  mere  specks  in  this  universe  of 
ether  we  realize  readily  from  the  statement  that  the 
radius  of  an  electron  is  probably  not  larger  than  2  X  10~13 
cm.  and  that  of  a  hydrogen  atom  is  about  2X1CT8  cm. 
The  whole  atom  is  perhaps  100,000  times  as  large  in 
diameter  as  the  electron.  In  other  words,  the  radius  of 
the  electron  is  about  as  large,  compared  to  the  radius 
of  the  atom,  as  is  the  radius  of  our  earth  as  compared 
to  the  radius  of  the  orbit  in  which  it  travels  around 
the  sun. 

We  may  consider  the  size  of  our  solar  system  to  be 
as  large  as  the  orbit  traveled  by  the  most  remote  planet. 


THE   ELECTRON 


91 


TABLE  I 

THE    ELEMENTS,    THEIR    ATOMIC    NUMBERS,    ATOMIC    WEIGHTS,     AND 
POSITIONS   IN   THE   PERIODIC   SERIES 

The  atomic  number  appears  at  the  left  of  the  symbol  for  each  element,  the  atomic 
weight  below ;  where  the  latter  are  not  in  the  order  of  the  atomic  numbers  they  are  in  italics. 

1  H 

1.008 


0 

I 

II 

III 

IV 

V 

VI 

VII 

VIII 

2  He 
3.99 

3  Li 
6.94 

4  Be 
9.1 

5B 
11.0 

6C 
12.00 

7N 

14.01 

8O 
16.00 

9F 
19.0 

10  Ne 
20.2 

11  Na 

23.00 

12  Mg 
24.32 

13  Al 
27.1 

14  Si 
28.3 

15  P 
31.04 

16  S 
32.06 

17  Cl 
35.46 

U  A 

19  K 

_..,  gj 

40.07 

21  Sc 
44.1 

22  Ti 
48.1 

23V 
51.0 

24  Cr 
52.0 

25  Mn 
54.93 

26  Fe  27  Co  28  Ni 
$534    5837    58.68 

29  Cu 

30  Zn 

31  Ga 

32  Ge 

33  As 

34  Se  ' 

35  Br 

63.57 

65.37 

69.9 

72.5 

74.96 

79.2 

79.92 

36  Kr 
82.92 

37  Rb 
B&45 

38  Sr 
87.63 

39  Y 

88.7 

40  Zr 
90.6 

41  Nb 
93.5 

42  Mo 
96.0 

4S- 

44  Ru  45  Rh  46  Pd 
101.7     102.9     106.7 

47  Ag 

48  Cd 

49  In 

50  Sn 

51  Sb 

52  Te 

53  I 

107.88 

112.40 

114.8 

118.7 

120.2 

1S7.5 

126.92 

130.2 

55  Cs 

:  -;J.M 

56  Ba 

\.  '>-.:>,- 

57  La 
139.0 

See 
Note 

73  Ta 
181.5 

74  W 
184.0 

75  — 

76  Os  77  Ir  78  Pt 
190.9  193.1   195.2 

79  Au 

EM  H_- 

81  Tl 

82  Pb 

83  Bi 

84  Po 

85  — 

197.2 

MNMJ 

204.0 

207.20 

208.0 

(210.0) 

86  Nt 

(2220) 

87  — 

88  Ra 
226.0 

89  Ac 
(227) 

90  Th 
232  15 

UrX2 

•':i4  ) 

92  Ur 
2382 



NOTE.  Each  vertical  column  forms  a  "group"  the  elements  of  which  have  the  valencies 
indicated  by  the  Roman  numerals  or  else  form  similar  compounds  with  other  elements  like 
hydrogen  and  oxygen.  Each  row  is  a  "period."  Corresponding  to  the  atomic  number  58 
is  the  rare  metallic  element  cerium.  For  simplicity  an  entire  period  of  rare  earths,  follow- 
ing cerium,  is  omitted  from  the  table  of  positions  although  they  are  included  in  the  list  of 
elements. 

THE  LIST  OF  ELEMENTS  IN  THE  ORDER  or  ATOMIC  NUMBERS: 


1  Hydrogen 
2  Helium 
3  Lithium 
4  Beryllium 
5  Boron 
6  Carbon 
7  Nitrogen 
8  Oxygen 
9  Fluorine 
10  Neon 
11  Sodium 
12  Magnesium 
13  Aluminium 
14  Silicon 
15  Phosphorus 
16  Sulphur 
17  Chlorine 
18  Argon 
19  Potassium 
20  Calcium 
21  Scandium 
22  Titanium 
23  Vanadium 

24  Chromium 
2.5  Manganese 
26  Iron 
27  Cobalt 
28  Nickel 
29  Copper 
30  Zinc 
31  Gallium 
32  Germanium 
33  Arsenic 
34  Selenium 
35  Bromine 
36  Krypton 
37  Rubidium 
38  Strontium 
39  Yttrium 
40  Zirconium 
41  Niobium 
42  Molybdenum 
43 

47  Silver 
48  Cadmium 
49  Indium 
50  Tin 
51  Antimony 
52  Tellurium 
53  Iodine 
54  Xenon 
55  Cesium 
56  Barium 
57  Lanthanum 
58  Cerium 
59  Praseodymium 
60  Neodymium 
61 

70  Ytterbium 
71  Lutecium 
7" 

73  Tantalum 
74  Tungsten 

*7~ 

76  Osmium 
77  Indium 
78  Platinum 
79  Gold 
80  Mercury 
81  Thallium 
82  Lead 
83  Bismuth 
84  Polonium 
85 

62  Samarium 
63  Europium 
64  Gadolinium 
65  Terbium 
66  Dysprosium 
67  Holmium 
68  Erbium 
69  Thulium 

86  Niton 
87 

88  Radium 
89  Actinium 
90  Thorium 
91  Uranium  X  2 
92  Uranium 

44  Ruthenium 
45  Rhodium 
46  Palladium 

92  THE   REALITIES  OF   MODERN  SCIENCE 

If  then,  as  there  seems  reason  to  believe,  the  hydrogen 
atom  consists  of  a  nucleus  and  one  electron  revolving 
about  it,  the  size  of  both  may  well  be  very  small  as 
compared  to  what  we  should  measure  as  the  size  of 
the  atom  or  of  its  inclosing  sphere.  Although  the  re- 
lations of  size  in  the  atom  are  much  like  those  of  our 
earth  and  its  orbit,  the  nucleus  is  very  much  smaller  in 
proportion  than  is  our  sun. 

In  the  case  of  atoms  there  are  two  ways,  which  we 
shall  discuss  more  fully  later,  in  which  electrons  may  be 
obtained.  In  the  first  case  we  may  jar  electrons  loose 
from  the  atom.  In  the  second  case  the  electron  may 
be  thrown  off  by  some  disturbance  which  has  its 
origin  in  the  atom  itself.  In  this  second  case  we  call 
the  substance  "radioactive."  Of  such  substances  ra- 
dium, discovered  in  1897  by  Monsieur  and  Madame 
Curie,  is  the  best  example.  Other  radioactive  sub- 
stances are  uranium,  thorium,  actinium,  and  polonium. 
The  atoms  of  such  substances  appear  to  be  disrupting ; 
not  that  all  of  them  do  so  at  once,  but  of  a  bit  of  such  a 
substance  some  of  the  atoms  are  always  breaking  down 
in  this  way.  In  such  a  breakdown  electrons  are  shot 
out  or  other  changes  take  place  and  the  result  is  new 
elements.1 

Uranium,  which  is  the  heaviest  known  atom,  has  at 
least  92  electrons,2  that  is,  it  has  92  electrons  exclusive 
of  any  which  may  be  contained  in  the  nucleus,  the  con- 
struction of  which  we  are  about  to  consider.  Between 

1  Radium  is  believed  to  be  a  product  due  to  the  disintegration  of 
uranium,  and  this  may  later  be  found  true  of  the  other  radioactive 
elements. 

2  The  determination  of  "atomic  numbers"  is  discussed  in  Chap- 
ter XXII. 


THE   ELECTRON  93 

hydrogen,  with  only  one  electron,  and  the  unstable 
uranium,  with  its  92,  lie  all  the  possible  elements.  If 
one  considers  that  an  atom  must  have  a  whole  number 
of  electrons  it  will  appear  that  the  other  84  known  ele- 
ments may  fit  in  between  hydrogen  and  uranium  and 
still  leave  six  yet  to  be  discovered.  The  names  of  the 
elements  with  their  numbers  of  electrons  are  given  in 
Table  I.  The  spaces  indicate  possible  elements  which 
have  not  yet  been  discovered.  These  may  some  day 
be  discovered  in  the  composition  of  some  distant  star 
or  perhaps  hi  some  rare  earth. 

What  is  the  character  of  the  nucleus  and  why  if 
electrons  repel  each  other,  are  they  not  all  dispelled 
and  all  atoms  disrupted?  The  answer  is  that  the 
character  of  the  nucleus  is  such  as  to  attract  towards 
itself  and  in  general  to  hold  within  the  atomic  radius 
the  various  electrons  of  the  atom.  We  may  say,  then, 
that  the  nucleus  is  itself  electricity,  but  of  a  different 
kind  from  that  of  the  electron.  For  convenience  and 
for  reasons  connected  with  the  history  of  the  science, 
we  call  the  electricity  of  the  electron  "negative"  and 
that  of  the  nucleus  "positive." 

The  names  were  introduced  by  Benjamin  Franklin 
about  1756,  long  before  the  electron  was  known,  for 
our  knowledge  of  the  latter  has  all  developed  since 
Rontgen's  discovery  in  1895  of  X-rays.1  In  Franklin's 
time  it  was  known  that  if  a  glass  rod  is  rubbed  with 
silk  it  is  electrified,"  that  is,  acquires  the  ability  to  at- 
tract light  particles,  as  bits  of  paper  or  pith  balls.  Two 

1  The  reader  probably  knows  most  of  X-rays  as  a  means  of  taking 
pictures  of  bones  or  teeth  for  surgical  diagnosis.  The  phenomenon 
is  discussed  further  in  Chapter  XIV. 


94  THE   REALITIES  OF   MODERN  SCIENCE 

pith  balls  electrified  by  contact  with  the  rod  are  found  to 
repel  each  other.  On  the  other  hand,  a  rod  of  sealing 
wax,  which  has  been  electrified  by  rubbing  it  with  cat's 
fur,  will  produce  effects  similar  to  those  of  the  glass 
rod.  But  there  is  an  important  difference.  A  pith  ball 
charged  by  the  glass  rod  and  one  charged  by  the  sealing 
wax  will  attract  each  other.  These  phenomena  may 
be  summarized  in  a  simple  law,  namely  :  like  electricities 
repel  and  unlike  attract.  To  the  electricity  of  the  glass 
rod  Franklin  gave  the  name  " positive"  and  to  that 
of  the  sealing  wax,  the  name  " negative."  However 
charges  of  electricity  are  produced  to-day,  we  still  use 
these  names. 

Before  returning  to  the  question  of  atomic  composi- 
tion we  shall  need  to  describe  further  the  phenomenon 
of  electrification.  It  will  be  found  by  test  that  the  silk 
is  negatively  charged,  when  the  glass  rod  is  positively 
charged.  Furthermore,  until  the  two  dissimilar  sub- 
stances are  brought  into  intimate  contact,  as  by  rub- 
bing, they  are  not  electrified.  The  normal  condition  of 
matter,  in  other  words,  gives  no  evidence  of  electricity. 
This  may  be  tested  by  presenting  the  silk  and  the  glass 
together  to  a  pith  ball,  which  has  been  charged,  and 
noting  that  the  latter  is  neither  repelled  nor  attracted. 
After  the  two  substances  are  electrified,  if  they  are  held 
close  together  and  tested  by  a  charged  pith  ball,  they 
will  produce  no  effect,  because  their  positive  and  neg- 
ative electricities  neutralize  in  their  effects.  Because 
of  this  equal  effect  we  conclude  that  unlike  electricities 
always  appear  in  equal  amounts. 

We  notice  also  from  such  an  experiment  that  when  a 
body  gives  no  evidence  of  electrification,  it  may  never- 


THE   ELECTRON  95 

theless  have  in  it  equal  and  unlike  electricities.  With 
this  in  mind  we  now  see  how  an  atom  or  a  molecule 
may  normally  appear  uncharged  and  yet  have  within 
it  equal  and  unlike  electricities. 

Recognizing,  then,  that  the  atom  is  composed  of  a 
nucleus  of  positive  electricity  and  of  electrons  we  may 
now  explain  the  electrification  which  is  produced 
by  rubbing  the  glass  with  silk.  During  the  intimate 
contact  some  of  the  electrons  which  are  not  very 
firmly  held  in  their  respective  atoms  become  attached 
to  other  atoms.  During  this  rearrangement1  of  the 
electrons  the  glass  loses  electrons  to  the  silk.  The  silk 
has  more  than  its  usual  number.  Conversely,  the 
glass  rod  has  less  than  its  normal  number  of  electrons 
and  therefore  appears  positively  charged.  In  other 
words,  for  the  glass  rod  there  are  not  sufficient  electrons 
to  neutralize  in  effect  the  positive  nuclei  of  its  atoms. 
The  positively  charged  glass  rod  will  attract  the  neg- 
ative silk  with  which  it  was  rubbed,  and  it  is  against  this 
attraction  that  we  do  work  in  separating  the  two  dis- 
similar substances  in  the  act  of  producing  electrification. 

We  may  also  ask  why  a  charged  glass  rod  attracts  an 
uncharged  pith  ball,  for  it  was  this  phenomenon  which 
first  called  attention  to  electricity.  The  mechanism  is 
simple.  The  rod,  being  positive,  draws  toward  itself 

1  We  may  imagine  a  very  simple  parallel  case  to  illustrate  how 
this  happens.  Consider  two  trains  which  meet  at  a  railroad  siding. 
Some  of  the  men  wander  about  from  car  to  car,  either  of  their  own 
train  or  of  the  other.  But  the  men  of  train  1  stick  more  closely  to 
then*  own  cars  than  do  those  of  train  2.  When  the  whistle  blows 
for  departure  the  first  train  will  have  more  and  the  second  train 
less  men  than  when  they  pulled  in.  If  we  picture  the  cars  as 
atoms  and  the  men  as  electrons  we  have  qualitatively  the  process 
of  "electrification  by  friction." 


96  THE  REALITIES  OF   MODERN  SCIENCE 

some  of  the  electrons  of  the  pith  ball.  The  side  of  the 
ball  near  the  rod  has  more  and  the  opposite  side  less 
than  its  normal  number  of  electrons.  The  nearer  side 
is  negative  and  the  more  distant  positive.  The  at- 
traction of  the  rod  for  the  nearer  side  is  greater  than  its 
repulsion  of  the  more  distant  side,  with  the  result  that 
the  pith  ball  is  urged  toward  the  rod. 

Returning  to  our  consideration  of  the  atom  we  may 
now  summarize  as  follows :  (1)  the  normal  state  of 
an  atom  is  uncharged ;  (2)  the  atom  consists  of  a  num- 
ber of  electrons,  and  a  nucleus  which  has  a  positive 
charge  just  equal  to  the  negative  electricity  of  the 
electrons;  (3)  the  operation  of  charging  a  body  with 
electricity  consists  in  causing  either  an  excess  or  a  de- 
ficiency in  the  number  of  electrons  in  the  body. 

The  nucleus  attracts  the  electrons  and  they  are  in 
general  very  firmly  held  in  the  atomic  radius.  But 
why  doesn't  the  attraction  which  exists  between  the 
positive  nucleus  and  the  negative  electrons  draw  them 
together?  To  this  question  the  scientist  has  as  yet 
been  unable  to  obtain  an  entirely  satisfactory  answer. 
For  this  and  other  questions  we  must  await  either  fur- 
ther theories  or  more  experimental  evidence.  The 
statements,  however,  which  we  have  so  far  made  as  to 
the  nature  of  the  atom  are  supported  by  experimental 
evidence  and  are  commonly  accepted. 

The  nucleus  we  recognize  as  positive  electricity,  but 
we  mean  thereby  merely  that  it  has  an  excess  of  pos- 
itive electricity  over  any  negative  which  it  may  also 
include.  In  fact  there  is  evidence  that  the  nucleus 
itself  contains  some  electrons.  In  the  case  of  radio- 
active substances  we  find  that  electrons  are  shot  off  and 


THE  ELECTRON  97 

also  atoms  which  we  recognize  to  be  helium.  Helium 
is  a  light  monatomic  gas,  twice  as  heavy  as  diatomic 
hydrogen.  Its  atom  consists  of  a  nucleus  and  two 
electrons. 

Since  such  an  atom  is  shot  off  from  radium  it  ap- 
pears to  mean  that  the  nucleus  of  the  radium  atom 
contains,  or  possibly  is  made  up  of,  helium  atoms.  It  is 
also  possible  that  these  helium  atoms  are  themselves 
made  up  of  hydrogen  atoms.  If  this  is  the  case,  the 
nucleus  of  each  helium  atom  should  consist  of  four 
nuclei,  like  that  of  the  hydrogen  atom,  and  two  elec- 
trons. This  helium  nucleus  would  then  require  two 
external  electrons  in  order  to  be  neutral.  That  the 
helium  atom  has  two  electrons  external  to  the  nucleus, 
we  know,  but  we  are  as  yet  in  the  dark  as  to  whether 
or  not  the  nucleus  is  formed  as  suggested  above.  The 
point  to  be  noted  is  that,  even  though  scientists  are  not 
yet  able  to  speak  definitely  of  the  composition  of  the 
nucleus,  there  is  sufficient  evidence  to  warrant  our 
bearing  in  mind  the  possibility  that  the  nuclei  of  all 
the  atoms  may  some  day  be  found  to  be  formed  by  the 
combination  of  electrons  and  a  number  of  positive  ele- 
mental charges l  all  just  like  the  positive  nucleus  of  the 
hydrogen  atom. 

1  If  we  use  the  term  "electron"  to  mean  an  element  of  electricity 
we  must  distinguish  between  this  positive  electron  and  the  negative 
electron  with  which  we  have  previously  dealt.  This  is  the  practice 
of  such  pioneers  in  electron  physics  as  J.  J.  Thomson,  Millikan, 
and  Rutherford.  Unfortunately  the  Century  Dictionary  has  not  re- 
corded this  usage.  It  happens,  however,  that  our  further  discus- 
sions concern  only  the  negative  electron,  and  for  convenience  we 
shall  continue  to  use  the  term  without  qualification  to  represent 
the  elementary  negative  charge. 
H 


98  THE   REALITIES   OF   MODERN   SCIENCE 

The  question  of  how  the  electrons  of  an  atom  group 
themselves  under  the  action  of  their  mutually  repellent 
forces  and  the  attraction  of  the  nucleus  is  a  difficult 
problem  of  mathematical  physics.  An  interesting 
study  has  been  made,  however,  of  the  manner  in  which 
a  number  of  similar  particles,  which  repel  each  other, 
would  group  themselves  in  a  plane,  under  the  action 
of  a  centrally  located  source  of  attraction.  A  number 
of  small  magnetic  needles  were  mounted  vertically 
through  small  corks  and  thus  floated  on  the  surface  of 
a  vessel  of  water.  Similar  poles  of  the  needles  pro- 
jected upward  through  the  corks.  A  long  magnet  was 
suspended  vertically  above  the  vessel  so  that  a  pole 
of  opposite  character  to  those  of  the  needles  might  act 
as  a  center  of  attraction.  It  was  found  that  the  con- 
figurations into  which  the  floating  poles  grouped  them- 
selves depended  upon  their  number.  Thus  three  poles 
placed  themselves  at  the  corners  of  an  equilateral  tri- 
angle, the  center  of  which  was  just  below  the  attracting 
magnet.  Similarly  four  magnets  group  at  the  corners  of 
a  square  and  five  at  the  vertices  of  a  regular  pentagon. 

When  six  magnets  enter  into  the  configuration  one 
goes  to  the  center,  immediately  below  the  pole  of  the 
large  magnet,  the  other  five  forming  a  pentagon.  With 
seven  and  eight  similar  figures  are  formed  with  one 
magnet  at  the  center.  With  nine  there  are  two  in  the 
center  and  seven  in  the  outside  ring.  With  ten  or 
eleven  there  is  an  inner  triangle  and  an  outer  ring  of 
either  seven  or  eight.  The  number  in  the  center  in- 
creases until,  when  the  total  is  fourteen,  there  is  a  pen- 
tagon inside  and  a  ring  of  nine  outside.  In  other  words, 
as  the  number  entering  into  the  configuration  is  in- 


THE  ELECTRON  99 

creased,  certain  arrangements  recur.  When  the  num- 
ber is  increased  to  fifteen  there  is  an  outer  ring  of  nine, 
an  inside  ring  of  five,  and  in  the  center  of  this  a  single 
magnet.  As  the  number  of  magnets  increases  certain 
configurations  appear  more  or  less  periodically,  that  is, 
at  definite  intervals. 

It  is  reasonable,  therefore,  to  expect,  as  prominent 
scientists 1  have  done,  that  whatever  the  positive 
nucleus  of  the  atom  may  be,  atoms  differing  in  their 
compositions  by  definite  numbers  of  electrons  may  in 
part  have  similar  configurations  for  their  electrons. 
In  so  far  as  the  electron  configuration  as  well  as  the 
number  determines  the  characteristics,  such  atoms 
should  have  certain  similarities  in  chemical  properties. 
If  a  list  is  made  of  atoms  in  the  order  of  the  number  of 
electrons  they  contain,  those  of  similar  properties  might 
be  expected  to  occur  periodically  in  the  list. 

To  some  extent  this  is  reasoning  after  the  fact.  That 
there  is  such  a  periodicity  to  a  list  of  the  elements, 
arranged  in  order  of  their  atomic  weights,  was  noticed 
years  before  the  electron  was  discovered.  Also,  some 
chemical  elements  are  similar  in  their  properties,  e.g. 
lithium  and  sodium  are  both  soft  white  metals.  They 
form  chlorides,  LiCl  and  NaCl,  respectively,  which 
have  quite  similar  properties.  Potassium  is  also  very 
similar  to  these  two  elements.  We  find  such  a  similar- 
ity between  the  compounds  of  sodium  and  potassium ; 
thus  we  are  all  familiar  with  the  similarity  of  caustic 
soda  and  caustic  potash,  NaOH  and  KOH,  respectively. 
The  atomic  weights  of  lithium,  sodium,  and  potassium 

1  Cf .  J.  J.  Thomson,  "The  Corpuscular  Theory  of  Matter," 
Chap.  VI,  Charles  Scribner's  Sons,  1907. 


100         THE   REALITIES   OF   MODERN   SCIENCE 

are  7,  23,  and  39,  respectively ;  hence  the  difference 
between  sodium  and  lithium  is  the  same  as  that  between 
potassium  and  sodium.  (Table  I  shows  equal  differences 
in  the  number  of  electrons.) 

The  classification  of  elements  according  to  atomic 
weights  was  first  worked  out  by  Newlands  in  1864.  It 
was,  however,  much  extended  and  elaborated  by  Men- 
delejeff  in  1869.  Finding  in  one  of  the  periods  a  place 
where  an  element  was  apparently  missing,  Mendelejeff 
predicted  that  an  element  of  about  a  certain  atomic 
weight,  with  a  certain  valence,  which  should  form 
certain  compounds  typical  of  its  group,  might  later  be 
discovered.  He  made  two  other  predictions  of  this 
character.  The  three  elements,  gallium,  scandium, 
and  germanium,  were  discovered  during  the  next  twenty 
years  and  found  to  have  essentially  all  the  properties 
which  had  been  predicted  for  them. 

The  characteristics  of  the  atoms  of  the  elements 
are  believed  to-day  to  be  due  to  the  number  and  con- 
figuration of  the  electrons.  The  power  of  combination 
is  undoubtedly  to  be  explained  in  terms  of  the  forces  ex- 
isting between  the  electrical  charges  of  atoms  which  are 
brought  close  together.  Although  positive  and  nega- 
tive electricities  in  the  atoms  are  equal,  their  distribu- 
tion might  well  be  such  that  a  net  force  would  act  be- 
tween two  dissimilar  atoms  if  they  were  very  close 
together.  The  cause  of  attraction  would  then  be  some- 
what similar  to  that  discussed  on  page  95,  the  result- 
ant force  being  due  to  the  fact  that  the  attracting 
electricities  of  the  two  atoms  were  on  the  average  closer 
to  each  other  than  were  the  repelling  electricities. 


CHAPTER  IX 
ENERGY 

IN  the  preceding  chapters  certain  facts  as  to  the 
composition  of  matter  have  been  stated,  which  may  be 
summarized  in  the  statement  that  matter  is  granular 
in  structure  and  electrical  in  nature.  In  all  matter 
we  have  reason  to  believe  that  the  constituent  parts 
are  in  motion.  By  the  motion  of  electrons  the  phe- 
nomenon of  contact  electrification  (cf .  page  95)  was  ex- 
plained. In  later  chapters  we  shall  discuss  further  the 
motions  of  the  electronic  constituents  and  also  of  the 
atomic  aggregates  which  compose  the  molecules  of 
matter.  Of  the  motions  of  the  molecules  themselves, 
at  least  in  the  case  of  liquids  and  gases,  there  is  familiar 
evidence  in  the  phenomenon  of  diffusion,  that  is,  the 
unaided  mixing  of  two  different  substances,  of  which 
the  diffusion  of  an  odor  through  air  is  an  example. 

Because  of  the  characteristic  of  inertia  (cf.  page  68) 
there  is  associated  with  all  the  motions  of  these  con- 
stituents of  matter  an  ability  to  do  work,  which  we  call 
energy.  To  say  that  all  bodies  in  the  universe  have 
inertia  is  to  say,  in  effect,  that  all  moving  bodies 
possess  energy,  but  offers  no  explanation  of  the  original 
cause,  the  energy  source  of  the  universe.  All  the 
physical  explanations  of  the  origin  of  our  earth,  or  our 
astronomical  system,  must  start  from  an  assumed  con- 

101 


102-        THE  MA&iTtf89*.6&  MODERN  SCIENCE 

dition,  or  reach  a  possible  initial  condition,  in  which 
there  was  present  a  sufficient  amount  of  energy  to 
account  for  the  present  distribution  of  energy  among 
the  component  parts. 

Although  in  this  chapter  we  shall  state  the  funda- 
mental ideas  as  to  energy  in  connection  with  visible 
and  tangible  bodies,  e.g.  stones  or  a>  baseball,  this 
should  not  mislead  as  to  the  relative  importance  of 
the  energy  associated  with  groups  of  molecules  as 
compared  to  that  of  the  molecules  themselves  or  their 
component  parts.  The  energy  which  is  most  important 
in  nature  is  not  that  of  bodies  like  trains  and  bullets, 
hammers  and  fists,  but  the  interatomic  and  molecular 
energy.  The  interatomic  energy  of  the  sun,  trans- 
mitted to  the  earth,  is  recognized  as  heat  and  light, 
and  is  the  cause  of  the  chemical  synthesis  in  growing 
plants.  It  is  interatomic  energy  which  causes  the 
heat  evolved  by  some  chemical  changes,  as  that  of  com- 
bustion. (The  heat  itself  we  shall  find  to  be  molecu- 
lar energy  of  the  body  which  is  heated.)  The  energy 
of  organic  life  is  the  result  of  innumerable  small  contri- 
butions of  interatomic  energy  and  to  some  extent  of 
molecular  energy.  The  rise  of  sap  in  plants  is  a  phe- 
nomenon of  molecular  energy. 

As  has  been  indicated  above,  energy  may  be  asso- 
ciated with  the  component  parts  of  the  atom,  with  the 
molecules  and  atoms  themselves,  or  with  bodies  of  more 
than  molecular  size.  Of  the  kinds  of  energy  we  dis- 
tinguish, however,  only  two,  namely  "potential"  and 
"kinetic,"  the  latter  due  to  motion  and  the  former 
existing  only  in  possibility.  Whenever  a  body  is 
given  the  ability  to  do  work  as  a  result  of  the  motion 


ENERGY  103 

which  is  communicated  to  it,  it  is  said  to  possess  kinetic 
energy.  Of  this  a  moving  bullet,  the  falling  weight  of  a 
pile  driver,  and  the  rotating  flywheel  of  an  automobile 
engine  are  examples.  In  the  same  way  moving  mole- 
cules or  the  moving  electrons  of  their  atoms  possess 
kinetic  energy. 

Kinetic  energy  is  the  only  kind  of  which  we  may  be 
conscious,  for  there  is  no  motion  in  the  case  of  poten- 
tial energy.  In  the  wood  beside  the  fireplace  and  the 
oxygen  in  the  room  there  exists  a  possible  source  of 
energy,  the  potential  energy  of  separation  of  two  chem- 
ical compounds,  which  is  only  manifested  when  they 
are  allowed  to  unite.  In  the  raised  weight  of  a  pile 
driver,  in  the  coiled  spring  of  a  watch,  in  the  powder 
of  a  cartridge,  in  the  water  of  lakes  and  reservoirs 
which  are  above  the  surrounding  ground,  there  is 
potential  energy. 

The  weight  of  a  pile  driver  possesses  potential  energy 
because  it  may  fall  toward  the  earth,  and  hence  it  is 
not  the  weight  itself,  but  rather  the  system,  of  the  weight 
and  the  earth  which  attracts  it,  which  possesses  this 
energy.  In  the  same  way  the  water  of  an  elevated 
reservoir  does  not  itself  possess  the  potential  energy, 
but  rather  the  system  of  which  it  is  a  part.  Potential 
energy  is  due  to  the  position  or  separation  of  the  parts 
of  a  system.  In  the  case  of  the  explosive  mixture  of 
the  cartridge  this  is  also  true,  and  the  energy  is  re- 
leased by  allowing  the  combination  of  the  separated 
parts  to  take  place. 

But  where  is  the  separation  in  the  case  of  the  coiled 
watch  spring?  If  anything,  the  parts  appear  to  be 
closer  together  when  the  spring  is  wound  than  when  it 


104         THE   REALITIES   OF   MODERN   SCIENCE 

is  unwound.  The  uncoiled  position  is  normal  for  the 
spring,  since  to  wind  it  requires  work.  In  this  condi- 
tion the  various  molecules  of  which  it  is  composed  have 
definite  positions  with  reference  to  each  other.  When, 
however,  we  bend  the  spring,  as  in  winding,  we  force 
them  to  assume  a  new  configuration.  Now,  whether 
in  the  act  Of  bending  we  pull  two  adjacent  molecules 
farther  apart  or  push  them  closer  together  we  change 
their  separations  and  potential  energy.  The  proof, 
however,  is  not  in  the  separation  but  in  the  subsequent 
ability  of  the  system  to  do  work. 

In  the  case  of  the  spring  the  system  is  molecular. 
In  that  of  the  explosive  mixture  in  the  cartridge  it 
consists  of  a  large  number  of  atoms,  grouped  into 
molecules  of  two  or  more  chemical  compounds.  The 
atoms  which  form  these  compounds  are,  of  course, 
capable  of  forming  other  compounds,  the  products 
of  the  explosion.  (If  the  explosive  is  a  single  com- 
pound it  must  be  an  unstable  one  from  which  the  com- 
ponent atoms  form  more  stable  compounds.)  In  the 
case  of  the  pile  driver  and  the  earth,  we  have  recog- 
nized a  separation  of  the  parts,  not  of  individual  mole- 
cules or  atoms,  but  of  molecules  in  the  bulk.  The  bulk 
of  molecules  forming  the  weight  are  separated  from 
the  bulk  of  those  which  form  the  earth. 

These  are  illustrations  not  only  of  the  energy  of 
separation,  but  of  the  three  different  types  of  separa- 
tion. In  the  system  of  pile  driver  and  earth  the 
separation  is  of  more  than  molecular  magnitude,  that 
is,  too  large  for  forces  between  molecules  to  come  into 
play.  In  the  coiled  spring  the  separations  are  small 
enough  for  molecules  to  exert  forces  upon  each  other. 


ENERGY  105 

In  the  explosive  mixture  the  separations  are  even  smaller 
in  size  and  concern  the  electrons  and  the  nuclei  of 
the  atoms,  which  come  together  into  new  combinations 
when  explosion  occurs. 

Potential  energy,  then,  is  the  ability  of  a  system  to  do 
work  as  the  result  of  the  displacement  of  its  parts, 
while  kinetic  energy  is  its  ability  as  a  result  of  their 
motions.  If  the  motions  are  those  of  whole  atoms 
or  molecules  the  energy  is  molecular  kinetic  energy. 
When  we  realize  that  every  moving  electron,  atom, 
or  molecule  in  the  universe  possesses  kinetic  energy, 
and  that  wherever  there  are  separations  which  tend  to 
change  there  is  potential  energy,  we  recognize  what 
an  enormous  store  of  energy  there  is  in  the  universe 
in  which  we  live.  The  questions  this  realization  sug- 
gests are :  How  much  of  this  energy  is  available  ?  and 
How  may  the  available  energy  be  utilized? 

To  obtain  the  idea  which  is  involved  in  the  word 
" available"  let  us  consider  the  case  of  the  attraction 
of  the  earth  for  parts  of  itself.  If  we  lift  a  shovelful 
of  earth  from  the  surface  we  do  so  against  the  attrac- 
tion of  the  rest  of  the  earth.  In  this  separation  work 
is  done  and  potential  energy  is  given  to  the  system 
composed  of  the  earth  and  the  shovelful,  which  we  are 
holding  apart.  As  another  example  note  that  each 
particle  of  a  mountain  forms  with  the  rest  of  the  earth 
a  system  with  potential  energy,  for  it  is  only  necessary 
to  start  a  stone  to  have  it  roll  crashing  down  the  moun- 
tain side.  But  suppose  that  we  stand  on  relatively 
flat  country  beside  a  deep  well  or  a  mine  shaft.  If 
we  start  a  stone  it  will  fall  down  the  shaft,  for  in  this 
case  also  the  system  possesses  potential  energy.  In 


106         THE  REALITIES  OF   MODERN  SCIENCE 

both  cases  the  energy  is  available,  for  it  is  possible  to 
decrease  the  separation,  since  in  both  cases  the  stone  has 
some  place  to  which  it  may  fall.  But  if  the  mine  shaft 
is  filled  up  so  that  the  stone  has  no  place  to  fall  the 
conditions  are  not  different  as  to  the  energy  which  the 
system  possesses,  for  there  is  still  the  same  separation 
existing  between  the  attracting  bodies.  As  to  avail- 
ability, however,  the  conditions  are  different,  for  the 
energy  is  no  longer  available. 

Consider  now  the  energy  of  the  system  which  the 
water  of  the  ocean  forms  with  the  earth.  All  the  drops 
which  have  the  same  separation  from  the  center  of 
the  earth  form  with  the  latter  systems  having  the  same 
potential  energy.  The  energy,  however,  of  all  these 
systems  is  not  available.  A  drop  of  water  on  a  cliff 
above  the  ocean  possesses  more  potential  energy, 
for  it  may  fall  to  the  level  of  the  surface  of  the  sea. 
Of  its  total  potential  energy  a  part  is  obviously  avail- 
able, and  this  part  is  the  difference  between  the  energy 
it  has  on  the  cliff  and  the  energy  which  any  drop 
has  on  the  surface  of  the  ocean.  Since  the  amount  of 
the  energy  which  is  available  depends  upon  the  dis- 
tance above  sea  level,  we  might  speak  of  water  at  sea 
level  as  having  zero  available  potential  energy  and 
measure  the  available  potential  energy  of  water  at 
higher  levels  by  the  distance.  We  might  further  speak 
of  sea  level  as  being  the  level  of  zero  gravitational 
potential,  allowing  the  word  "  energy  "  to  be  understood. 

Let  us  follow  this  idea  a  little  further  and  see  where 
it  leads.  We  did  not  call  attention  to  the  drops  of 
water  but  rather  to  their  " locus,"  namely  the  surface 
of  the  sea.  We  spoke,  in  other  words,  of  a  locus  of 


ENERGY  107 

zero  potential.  Since  the  potential  energy  of  a  drop 
of  water  depends  upon  where  it  is,  that  is,  upon  the 
point  which  we  are  considering,  we  are  justified  in 
speaking  of  the  gravitational  potential  of  any  point 
as  meaning  the  available  potential  energy  of  a  drop  of 
water  placed  at  that  point.  All  the  points  about  the 
earth  where  such  a  drop  of  water  would  have  the  same 
potential  energy  form  an  equipotential  surface.  For 
example,  if  we  consider  a  point  a  foot  above  sea  level 
it  will  have  a  certain  potential,  but  every  other  point 
which  is  one  foot  above  sea  level  will  have  the  same 
potential.  Through  these  we  may  think  of  a  surface 
as  existing,  not  a  real  surface  like  that  of  the  sea  itself, 
but  an  abstract  one  like  those  which  are  considered  in 
geometry.  Similarly,  other  equipotential  surfaces  may 
be  imagined. 

This  idea  of  potential  is  of  peculiar  value  in  the 
study  of  electricity,  but  we  are  limiting  our  present 
discussion  to  gravitational  potential.  Although  this 
terminology  is  largely  used  only  in  such  study,  equi- 
potential surfaces  are  frequently  dealt  with.  The 
surveyor,  locating  a  railroad  route,  endeavors  as  far 
as  possible  to  find  it  on  an  equipotential  surface.  He 
does  not  call  it  that,  for  he  usually  says  he  wishes  to 
obtain  a  level  line  or  else  a  line  of  low  grades.  In 
passing  from  a  point  of  lower  to  one  of  higher  poten- 
tial, that  is,  from  one  nearer  to  the  center  of  the  earth 
to  one  farther  therefrom,  work  must  be  done  in  moving 
a  body ;  and  as  far  as  possible  the  engineer  wishes  the 
only  work  to  be  that  required  to  overcome  the  friction 
of  the  moving  train.  Of  course,  if  his  road  was  going 
to  carry  freight  only  one  way,  he  would  seek  for  a 


108         THE   REALITIES   OF   MODERN  SCIENCE 

route  which  led  from  a  high  potential  to  a  low  poten- 
tial. 

Returning  to  our  idea  of  the  gravitational  poten- 
tial of  a  point  as  the  potential  energy  of  a  drop  of  water 
at  that  point,  the  reader  has  doubtless  felt  that  the  drop 
of  water  forms  an  unsatisfactory  unit.  With  this  we 
agree  and  suggest  that  we  speak  of  a  pound  of  water 
or  preferably  a  gram  of  water,  and  then  abstracting 
the  water  leave  merely  the  mass.  The  gravitational 
potential  of  a  point  is  the  potential  energy  of  a  gram 
at  the  point. 

In  case  we  use  grams  we  should  use  the  centimeter 
as  the  unit  of  distance.  The  potential  of  the  surface 
which  is  one  centimeter  above  sea  level  we  shall  call 
one  gram-centimeter  (1  g.  cm.).  The  amount  of  work, 
then,  which  one  gram  can  do  in  falling  from  a  point 
one  centimeter  above  sea  level  to  sea  level  is  a  gram- 
centimeter.  Conversely,  it  would  require  one  gram- 
centimeter  to  lift  a  gram  from  sea  level  to  a  point  one 
centimeter  above  it. 

Whenever  we  pass  from  a  point  of  lower  potential 
to  one  of  higher  potential,  work  is  required,  that  is, 
energy  must  be  supplied.  Similarly,  in  passing  from 
a  point  of  higher  to  one  of  lower  potential,  energy  is 
released.  When  we  supply  energy  we  do  work,  that 
is,  we  exert  a  force  through  a  distance.  On  the  other 
hand,  when  energy  is  released,  work  is  done  for  us. 
In  both  cases  a  force  comes  into  play  when  the  poten- 
tial is  changed.  If  the  potential  is  increased  a  force 
must  be  applied  to  the  system.  On  the  other  hand, 
if  the  potential  is  decreased  the  acting  force  is  supplied 
by  the  system  under  consideration.  In  a  later  chapter 


ENERGY  109 

we  shall  define  " force"  quantitatively  in  terms  of  the 
change  in  energy. 

When  we  move  a  body  from  a  point  of  lower  to  one 
of  higher  gravitational  potential  we  must  supply  a 
force  and  we  do  work.  Potential  energy  is  thus 
stored  in  the  system.  But  when  a  body  is  allowed  to 
pass  from  a  point  of  higher  to  one  of  lower  potential, 
e.g.  to  fall,  what  becomes  of  the  potential  energy 
which  it  possessed?  We  know  that  the  body  moves 
faster  and  faster  as  it  falls.  An  acceleration  is  the  result 
when  its  potential  energy  is  allowed  to  decrease.  We 
shall  postpone  the  quantitative  definition  of  accelera- 
tion to  the  chapter  on  " Rates"  and  be  content  at  this 
point  with  the  qualitative  definition  implied  above. 
The  very  fact  that  the  body  falls,  that  is,  is  set  into 
motion,  means  that  it  is  given  kinetic  energy.  The 
farther  it  falls  the  more  potential  energy  the  system 
has  expended.  But  the  farther  it  falls  the  greater  is  its 
speed,  the  larger  its  kinetic  energy,  and  the  smaller  its 
potential  energy. 

Suppose  we  consider  a  numerical  case  of  a  body  of 
one  pound  lifted  20  feet  above  sea  level.  It  has  a 
potential  energy  of  20  ft.  Ibs.  Of  course,  if  the  body 
was  two  pounds  it  would  have  twice  as  much  energy. 
In  fact  the  potential  of  the  point  to  which  it  is  lifted 
is  the  potential  energy  possessed  by  each  pound  of 
the  body,  or  as  we  say  the  potential  of  this  point  is 
the  energy  per  pound  of  a  body  placed  there.  Allow 
the  body  to  fall.  When  it  is  at  a  point  19  feet  above 
the  zero  of  potential  it  has  19  ft.  Ibs.  of  potential  per 
pound.  How  much  kinetic  energy  has  it?  If  none 
of  the  energy  of  the  system  has  been  given  up  to  other 


110         THE   REALITIES  OF   MODERN  SCIENCE 

systems,  as  for  example,  to  the  molecules  of  the  air 
through  which  the  body  falls,  then  its  kinetic  energy  is 
1  ft.  Ib.  for  each  pound  of  the  body.  When  it  has  fallen 
to  a  point  18  feet  above  the  zero  of  potential,  it  has  2 
ft.  Ibs.  of  kinetic  energy  for  each  pound  of  the  body. 
The  decrease  in  potential  energy  is  always  equal  to  the 
increase  in  kinetic  energy,  provided  that  none  of  the 
available  energy  is  allowed  to  get  away  from  the  system. 
There  is  a  change  in  the  kind  of  energy  going  on  steadily 
as  the  body  falls,  but  no  change  in  the  total  amount 
of  energy  possessed  by  the  system.  Such  a  system, 
where  energy  is  neither  added  nor  subtracted  by 
some  outside  system,  is  called  "conservative." 

In  practice,  however,  we  do  not  find  conservative 
systems.  All  the  systems  with  which  we  have  to  do 
either  lose  or  gain  energy  from  other  systems.  In  the 
case  of  a  body  falling  through  air  some  of  the  energy 
is  imparted  to  the  molecules  of  the  air  which  the 
body  pushes  aside  in  falling.  These  air  molecules  are 
set  into  motion  ;  that  is,  they  are  given  kinetic  energy. 
The  energy  given  to  them  is  subtracted  from  the  sys- 
tem which  we  are  considering.  The  system  of  earth 
and  body  is  not,  then,  a  conservative  system.  If, 
however,  we  consider  the  system  to  be  composed  not 
only  of  the  earth  and  the  body  but  also  of  the  molecules 
of  the  air  we  do  have  a  conservative  system. 

In  the  story  of  Galileo's  experiment  we  saw  that  a 
falling  body  does  not  fall  as  quickly  if  there  is  fric- 
tion. We  now  see  from  our  knowledge  of  energy  why 
this  is  so.  In  falling  through  any  given  distance  a 
certain  definite  amount  of  energy  is  released.  If  all 
this  energy  is  available  for  accelerating  the  body,  it 


ENERGY  111 

will  acquire  a  greater  speed  and  hence  take  less  time  to 
fall  than  if  part  of  the  energy  must  be  expended  in 
accelerating  the  molecules  of  air  which  are  adjacent 
to  its  path. 

The  greater  the  amount  of  energy  which  friction 
subtracts  from  the  system,  the  smaller  the  kinetic 
energy  imparted  to  the  body  and  hence  the  more  slowly 
does  it  move.  Now,  it  is  possible  to  arrange  the  fric- 
tion which  a  falling  body  must  overcome  so  as  to  sub- 
tract all  of  its  energy.  When  this  is  done,  the  body, 
of  course,  descends  so  slowly  and  uniformly  that  it  is 
scarcely  moving  when  it  reaches  its  final  resting  place. 
We  accomplish  this  every  day  without  realizing  the 
physics  of  it.  When  one  sets  a  glass  of  water  on  a  table 
he  does  so  in  such  a  way  that  it  doesn't  bump ; 
that  is,  he  makes  sure  that  it  has  no  kinetic  energy, 
retarding  its  fall  by  absorbing  into  his  own  muscles 
the  energy  which  is  released  as  the  glass  descends  from 
a  point  of  higher  to  one  of  lower  potential. 

Perhaps  the  most  striking  illustration,  however,  is 
the  parachute  which  was  used  in  the  War  for  escaping 
from  observation  balloons.  This  is  merely  a  huge 
silk  umbrella  without  ribs.  In  the  top  is  a  small  hole 
through  which  the  air  can  escape  slowly  and  with 
considerable  friction.  When  the  parachute  is  released 
from  the  balloon  it  falls  very  rapidly,  until  the  air 
has  opened  it.  Thereafter  its  descent  is  quite  gradual. 
When  the  parachute  is  working  properly  the  bal- 
loonist arrives  at  the  earth  without  an  unfortunate 
amount  of  kinetic  energy,  for  the  energy  which  he 
possessed  at  the  higher  altitude  has  been  entirely  con- 
verted into  molecular  kinetic  energy  of  the  air. 


112         THE  REALITIES  OF   MODERN  SCIENCE 


What,  however,  is  the  effect  of  increasing  the  kinetic 
energy  of  the  molecules  of  a  substance?  An  experi- 
ment which  was  performed  by  the  Englishman,  Joule, 
in  about  1843  not  only  answered  this  question  but  is 
the  basis  of  the  reasoning  as  to  energy  which  we  are 
able  to  follow  to-day.  He  arranged  a  falling  weight 
so  that  it  turned  a  paddle  wheel  in  a  vessel  of  water 
as  shown  in  Fig.  8.  The  motion  of  the  falling  weight 

is  retarded  in  part  by 
the  friction  of  the  pad- 
dle wheels  and  water 
and  in  part  by  a  second 
weight  which  it  lifts. 
This  second  weight  was 
increased  until  its  re- 
tarding effect  plus  that 
of  the  paddles  was  just 
sufficient  to  insure  that 
the  first  weight  fell  so  gradually  as  practically  to  have 
no  kinetic  energy.  The  first  weight,  being  greater 
than  the  second,  lost  potential  energy  while  the  latter 
gained.  The  total  loss  in  potential  energy  of  the  system 
formed  by  the  earth  and  the  two  weights  was  the  same 
as  if  a  single  weight,  equal  to  the  difference  of  the 
two,  had  fallen  through  the  distance  through  which  the 
first  weight  fell. 

Joule  found  that  the  rapidly  revolving  paddle  wheels 
heated  the  water,  and  he  measured  the  rise  in  its 
temperature  with  a  thermometer.  He  made  several 
trials  and  he  found  that  for  every  42,700  g.  cm.  of 
potential  energy  which  the  weights  lost  there  was  an 
equivalent  rise  of  temperature  of  one  degree  Centi- 


FIG.  8. 


ENERGY  113 

grade  1  for  one  gram  of  the  water.  In  other  words, 
he  found  that  energy,  which  was  apparently  lost,  was 
not  lost,  as  had  been  believed  up  to  that  time,  but 
was  converted  into  molecular  energy  and  manifested 
as  heat.  He  also  found  the  numerical  relation  between 
temperature  and  the  molecular  energy  of  water.  He 
is  usually  said  to  have  determined  the  "  mechanical 
equivalent  of  heat."  This  term,  however,  is  mislead- 
ing, and  we  shall  discuss  it  later.2 

Joule's  experiments  extended  through  several  years 
and  covered  a  wide  range  of  possible  cases,  of  which  the 
one  described  above  is  merely  the  simplest  for  such  a 
discussion  as  ours.  In  all  cases  he  found  that  when 
energy,  which  was  associated  with  ponderable  bodies, 
disappeared  there  was  to  be  measured  a  perfectly 
definite  amount  of  increase  in  heat  in  the  parts  of  the 
system  with  which  he  was  working.  Of  course,  his 
laboratory  methods  were  not  always  as  precise  as  they 
might  be  made  to-day  ;  and  some  of  them  do  not  permit 
of  very  accurate  observations.  His  experiments,  how- 
ever, satisfied  the  scientific  world  of  the  truth  of  his 
idea  that  energy  is  never  destroyed  but  merely  changed 
in  kind  or  in  location. 

This  fundamental  fact  of  physical  science,  which 
Joule  demonstrated  is  known  to-day  as  the  Principle 
of  the  Conservation  of  Energy,  but  is  better  expressed 
as  the  Principle  of  the  Indestructibility  of  Energy. 

Energy  and  matter  are  both  indestructible.  Energy, 
as  we  have  seen,  may  change  its  form  and  its  location, 

1  One  degree  centigrade  is  1.8  degrees  on  the  Fahrenheit  scale  of 
temperatures.     Also  0°  C.  is  the  same  temperature  as  32°  F. 

2  Cf.  Chapter  XXI. 

i 


114         THE   REALITIES  OF   MODERN  SCIENCE 

but  its  amount  remains  unchanged.  Matter  may 
change  its  form,  being  solid,  liquid  or  aeriform.  We 
cannot,  however,  either  create  or  destroy  matter.  Nor 
are  there  any  processes  of  nature  whereby  it  may 
be  done.  It  may  undergo  transformations,  appear- 
ing to  us  now  as  one  chemical  substance  and  now  as 
another,  but  the  total  amount  of  matter  in  the  uni- 
verse remains  unaltered. 

Matter  and  Energy,  two  indestructibles  of  the  uni- 
verse, are  the  entities  in  terms  of  which  we  must  ex- 
plain all  physical  phenomena.  They  are  the  realities 
of  modern  science.  In  connection  with  the  first,  we 
must  always  bear  in  mind  the  granular  and  electrical 
composition.  In  connection  with  the  second,  we  shall 
need  a  further  law  as  to  availability.  Some  ideas  as 
to  this  have  been  given  in  this  chapter.  A  general 
statement  is  contained  in  what  is  known  as  the  ' l  Second 
Law  of  Thermodynamics."  Thermodynamics,  as  a 
division  of  science  dealing  with  the  forces  due  to  heat, 
was  established  before  scientists  appreciated  the  unity 
of  physical  science.  Its  first  law,  embodying  the  rela- 
tion which  was  discovered  by  Joule,  is  essentially  the 
principle  of  the  conservation  of  energy.  Its  second  law 
states  the  limits  of  the  availability  of  energy. 

To-day  we  realize  that  these  laws  are  not  limited 
in  their  application  to  a  particular  division  of  science 
but  are  fundamental.  In  fact,  it  would  be  preferable 
if  they  could  be  spoken  of  as  the  first  and  second  laws 
of  Energy,  but  we  must  use  the  names  which  were 
attached  to  them  at  the  time  they  were  formulated. 


CHAPTER  X 

SOME  USES  OF  MATHEMATICS 

WE  are  all  familiar  with  the  convenience  of  abbreviat- 
ing words  by  single  letters.  Certain  cases  are  found 
in  dictionaries  and  are  taught  in  schools  so  that  their 
use  is  practically  universal  within  the  limits  of  the 
particular  language.  The  letter  is  used  as  a  symbol 
for  the  word  just  as  the  word  is  itself  a  symbol  for  the 
idea.  The  convenience  is,  of  course,  one  of  brevity  in 
writing.  In  mathematics  symbols  are  used  which 
are  universally  recognized  without  regard  to  the  lan- 
guage of  the  student.  This  is  a  convenience,  but  the 
chief  advantage  which  mathematics  offers  to  science 
is  its  methods  of  handling  symbolized  ideas  so  as  to 
reach  conclusions  which  in  some  instances  would  be 
practically  unattainable  by  any  other  form  of  reason- 
ing. In  fact,  mathematics,  usually  defined  as  "the 
science  concerned  with  the  logical  deduction  of  con- 
sequences from  general  premises,"  may  be  concisely 
defined  as  "symbolized  logic."  Let  us  then  examine 
a  few  illustrations  which  will  show  this  convenience 
of  abbreviation  and  also  in  a  very  elementary  way 
the  logical  deduction  of  consequences  from  assumed 
premises. 

In  determining  the  area  of  a  rectangle  we  multiply 
the  length  by  the  breadth.  Thus  if  the  sides  are  3 
and  4  inches  we  say  the  area  is  12  square  inches  since 

115 


116         THE   REALITIES  OF   MODERN   SCIENCE 

3X4  =  12.  The  general  rule  for  finding  the  area 
would  be  as  follows  :  Measure  the  length  and  the 
breadth  in  the  same  units  (e.g.  both  in  inches)  and 
multiply  the  number  of  units  expressing  the  length 
by  the  number  expressing  the  breadth;  the  product 
will  be  the  number  of  times  the  square  unit  of  area 
is  contained  in  the  rectangle. 

Such  a  familar  rule  may  be  more  concisely  stated  in 
symbols  as  follows:  If  L,  J5,  and  A  express  the  length, 
breadth,  and  area  of  any  rectangle  then 

A=LB  (1) 

This  equation  states  the  relation  of  the  three  magni- 
tudes L,  B,  and  A.  Divide  both  sides  of  this  equality 
by  B  and  we  obtain 

L  =  j  (2) 

This  new  form  of  the  relation  indicates  what  of  course 
we  have  known  since  our  early  study  of  arithmetic, 
namely,  that  the  length  of  a  rectangle  is  to  be  found  by 
dividing  its  area  by  its  breadth.  Similarly  dividing 
both  sides  of  the  equality  (1)  by  L  gives 

B  =  y  (3) 

LI 

which  states  a  similar  relation  for  the  breadth  of  a 
rectangle. 

Equations  (1),  (2),  and  (3)  are  obviously  all  the  forms 
in  which  the  relationship  of  the  sides  and  area  of  a 
rectangle  may  be  explicitly  expressed.1  We  need  to 

T  7? 

1  The  relation  —  =  1  is  the  general  but  implicit  expression  of 

A. 

the  relationships  of  equations  (1),  (2),  and  (3). 


SOME   USES  OF  MATHEMATICS  117 

remember  but  one  of  them,  since  any  one  is  obtainable 
from  another  by  the  simplest  algebraic  process.  In 
passing  from  one  form  to  another  we  have  reasoned 
mathematically. 

In  the  case  of  any  law  of  science  which  may  be  ex- 
pressed in  the  form  Z  =  XY  where  X,  Y,  and  Z  are  the 
magnitudes  with  which  the  law  deals  we  may  always 

reason  by   the   same   mathematical   processes  as  we 

y 
employed  above  and  arrive  at  the  results  of  Jf  =  —  and 

17 

Y  =  — .     Now,  it  happens  that  in  an  ordinary  elemen- 
X 

tary  course  hi  physics,  such  as  that  of  a  high  school, 
the  student  meets  about  forty  physical  laws  which  are 
expressible  by  just  this  simple  relation.  In  our  dis- 
cussion we  have  arrived  quickly  at  the  general  and 
abstract  case.  Partly  because  of  the  immaturity  of 
high-school  students  and  partly  because  of  certain 
traditions  and  inhibitions  of  their  teachers  it  too  fre- 
quently happens  that  each  of  these  laws  is  dealt  with 
as  a  special  concrete  case.  The  result  is  that  the  simple 
mathematical  transformations  which  we  have  indicated 
above  occasion  what  is  perhaps  undue  difficulty,  and 
overemphasize  what  are  popularly  considered  the 
mathematical  difficulties  of  the  subject. 

There  are  two  reasons  why  this  typical  equation 
should  be  carefully  studied.  The  first  is  the  obvious 
one  that  it  would  make  the  subsequent  work  of  the 
student  much  easier,  for  whenever  he  meets  a  law 
expressible  hi  this  form  he  knows  that  he  may  apply 
the  same  mathematical  processes  of  reasoning  with 
similar  results.  The  second  advantage  is  that  he  may 


118     .    THE   REALITIES  OF    MODERN  SCIENCE 

then  concentrate  his  attention  more  on  the  physics 
of  the  relation  and  not  so  much  on  the  mathematics. 

In  other  words,  mathematics  to  the  scientist  or 
engineer  is  merely  a  tool,  which  he  should  use  like  a 
good  workman  almost  by  second  nature,  thinking  not 
of  the  tool  but  of  the  work  which  he  wishes  to  do.  Of 
course,  a  workman  selects  the  tool  required  for  the 
particular  job  which  he  has  in  mind,  but  he  does  so  as 
the  result  of  previous  knowledge  as  to  how  the  work 
is  to  be  done  and  of  previous  experience  with  his  tools. 
In  this  chapter  we  shall  consider  briefly  relations  sym- 
bolized by  equations  of  the  form  Z  =  XY,  which  proves 
so  troublesome  to  the  high-school  student,  and  in  the 
next  chapter  we  shall  consider  another  illustration  of 
mathematics  which  frequently  proves  unnecessarily 
difficult  to  his  college  brother. 

One  familiar  case  of  the  relationship  Z=XY  has 
been  illustrated  by  the  area  of  a  rectangle.  Let  us 
consider  another  in  the  expression  of  the  law  for  the 
volume  of  a  rectangular  box.  We  derive  the  law  for 
the  volume  by  noticing  that  if  the  area  of  the  base  is 
increased,  e.g.  doubled,  the  volume  is  proportionately 
increased,  e.g.  doubled.  Symbolizing  the  volume  by 
V  and  the  area  as  before,  we  express  this  fact  as 

V  oc  A  i  (4) 

Similarly,  if  the  area  of  the  base  is  kept  constant  and 
the  height  changed  the  volume  is  altered  proportion- 
ately ;  thus,  representing  height  by  H 

V  oc  H  (5) 

1  The  symbol  oc  obviously  means  "is  proportional  to." 


SOME   USES  OF  MATHEMATICS  119 

Now  these  two  facts  may  be  expressed  in  the  single 
relation 

V  oc  AH  (6) 

This  is  the  law  for  the  volume  of  a  rectangular 
parallelepiped.  It  is  not  yet  in  the  form  for  numerical 
calculation  because  we  have  not  yet  decided  upon  the 
units  hi  which  the  three  magnitudes,  V,  Hy  and  A  are 
to  be  measured  or  expressed. 

We  recognize  that  any  magnitude,  e.g.  an  amount  of 
money,  is  expressed  by  a  number,  a  "numeric/'  as  we 
say,  and  a  unit.  The  numeric  indicates  how  many 
times  the  chosen  unit  is  contained  in  the  given  magni- 
tude. The  greater  the  unit,  the  smaller,  of  course,  the 
corresponding  numeric.  Thus 

$3  =  300^,  that  is  3(1  dollar)  =300(1  cent) 
or  1  dollar,  300  _  100 
1  cent  " '  3         1 

In  the  expression  of  any  given  magnitude  the  numeric 
is  inversely  as  the  unit. 

The  numeric  expressing  the  volume  depends  upon 
the  choice  of  unit  for  measuring  volume.  Similarly, 
the  product  AH  will  depend  upon  the  units  in  which 
these  two  magnitudes  are  expressed.  Consider  for 
example  the  concrete  numerical  problem  of  a  corn- 
crib  of  area  10  sq.  ft.  and  height  4  ft.  For  this  case 
AH  is  obviously  40  cubic  feet.  But  if  we  measured 
the  volume  by  pouring  in  corn  we  should  find  it  to  be 
about  32  bushels.  That  is,  in  bushels  the  numeric 
of  the  volume,  V,  is  32.  By  selecting  the  unit  we  may 
make  the  numeric  anything  we  please.  It  would, 
however,  be  a  convenience  to  choose  the  unit  expressing 


120         THE  REALITIES   OF    MODERN  SCIENCE 

the  volume  so  that  the  corresponding  numeric  would 
be  the  same  as  that  expressing  the  product  AH. 

Of  course,  in  this  concrete  numerical  problem  we 
recognize  that  the  unit  we  should  use  is  the  cubic 
foot,  since  we  are  expressing  the  area  in  square  feet 
and  the  height  in  feet.  But  we  are  dealing  with  an 
evident  problem  so  that  the  method  which  we  are  em- 
ploying may  not  be  obscured  by  incidental  difficulties. 
Let  us  represent  the  desired  and  supposedly  unknown 
unit  for  volume  by  (v) ;  then,  since  the  numeric  corre- 
sponding to  AH  is  10X4  we  write 

40(0)  =  10(1  sq.  ft.)X4(lft.) 
or  0)  =  (1  sq.  ft.)(l  ft.)- 

That  is,  the  unit  of  volume  is  that  of  a  rectangular 
parallelepiped  for  which  the  base  is  one  unit  of  area 
(1  sq.  ft.)  and  the  height  is  one  unit  of  length  (1  ft.). 

So  far  our  reasoning  has  dealt  mostly  with  a  concrete 
numerical  example.  Concrete  numerical  illustration, 
without  an  expression  or  development  of  the  general 
principle  involved,  was  the  method  of  the  first  textbook 
of  science  of  which  we  know,  the  Ahmes  papyrus. 
This  was  the  work  of  an  Egyptian  priest  about  1700 
B.C.  and  was  based  upon  an  earlier  text  of  which  no 
portions  have  been  found,  which  antedated  it  by  500 
years  at  least.  The  manuscript  describes  itself  as 
"  Instructions  for  arriving  at  the  knowledge  of  all 
things,  and  of  things  obscure,  and  of  all  mysteries." 
If  one  is  satisfied  in  science  with  a  concrete  case  and 
does  not  go  on  to  abstract  the  general  principle,  his 
attitude  will  be  about  that  of  the  ancient  Egyptian. 
It  was  the  Greeks,  as  we  saw,  who  started  science 


SOME  USES  OF  MATHEMATICS  121 

about  600  B.C.,  by  their  ability  to  express  and  reason 
with  abstract  ideas.  If  one  catches  up  with  them  he 
is  only  about  400  years  behind  the  times  in  point  of 
view,  for  Greek  thought  dominated  such  scientific 
spirit  as  was  shown  in  the  medieval  ages,  until  Galileo 
in  the  17th  century  connected  theory  and  practice 
through  experimentation.  To  catch  up  with  the 
present-clay  trend  is  not,  however,  as  hard  a  task  as 
these  dates  would  seem  to  imply,  for,  if  we  accept  the 
inheritance,  we  are  heirs  to  the  successes,  not  the 
failures  and  trials,  of  our  scientific  forebears. 

Returning  to  our  concrete  problem  we  may  now  make 
a  general  statement.  If  a  law l  is  expressed  as  Z  oc  XY, 
then  Z  =  XY  if  the  units  (a;),  (y),  and  (z)  are  so  chosen 
that  they  are  related  as  (z)  =  (x)(y).  In  the  case  of  a 
rectangular  area,  if  the  sides  are  measured  in  feet 
(x)  =  1  ft.  and  (y)  =  1  ft.,  hence  (z)  =  (1  ft.)(l  ft.)  or  lit:2 
More  generally,  if  the  unit  of  length  which  we  adopt 
is  symbolized  by  (L)  the  unit  of  area  will  be  (L)2  and 
the  unit  of  volume  will  be  (L)3  e.g.  1  ft.3 

Starting  with  a  chosen  unit  of  length  we  derive,  as 
a  consequence  of  the  law  of  area,  a  unit  of  area  which 
is  expressed  in  terms  of  our  fundamental  unit.  Simi- 
larly, from  the  law  of  volume  we  derive  a  unit  of 
volume  which  is  expressed  in  terms  of  the  unit  of 
length. 

In  the  sense  of  being  derived  from  some  other  unit, 
the  unit  of  length  is  not  itself  a  derived  unit.  It  is  a 
fundamental  unit,  as  we  have  implied  above.  Such 

1  More  generally  if  Z&XY  then  Z  =  kXY,  where  k  is  a  factor  of 
proportionality.  This  is  the  defining  equation  of  Z  in  terms  of 
X  and  y,  and  k  depends  upon  the  choice  of  units. 


122         THE  REALITIES  OF   MODERN  SCIENCE 

units  have  all  been  more  or  less  arbitrarily  chosen. 
When  the  French,  at  the  time  of  their  Revolution, 
by  adopting  the  metric  units/  laid  the  basis  of  the 
present  system  of  derived  units,  for  the  measurements 
of  all  the  magnitudes  with  which  science  deals,  they  at- 
tempted to  obtain  a  unit  which  would  not  be  dependent 
on  an  arbitrarily  chosen  standard,  as  was  the  English 
' '  yard, ' '  and  would  have  a  physical  significance.  There 
was  thus  chosen  the  one  ten-millionth  part  of  the  dis- 
tance from  the  equator  to  the  pole  along  the  meridian 
passing  through  Paris.  Obviously,  the  determination 
of  this  distance  was  a  matter  of  astronomical  observa- 
tions rather  than  of  direct  measurement.  The  dis- 
tance as  finally  determined  was  marked  off  by  two 
fine  lines  on  a  long  platinum  bar.  This  distance  is 
the  "meter." 

As  a  matter  of  fact,  later  measurements  showed 
that  it  is  not  exactly  the  desired  fraction  of  the  earth's 
quadrant.  The  distance  is  nevertheless  the  accepted 
standard  of  length  for  all  scientific  as  well  as  for  many 
commercial  purposes. 

Even  if  the  meter  had  been  the  desired  fraction  of 
the  earth's  quadrant  it  would  not  have  been  a  derived 
unit  but  merely  a  submultiple  of  an  arbitrarily  chosen 
length.  The  choice  of  a  ten-millionth  as  the  sub- 
multiple  made  a  unit  of  convenient  size,  a  matter  of 
39.37  inches  or  about  10  per  cent  more  than  our  familiar 
unit  of  the  yard.  Of  course,  one  idea  in  basing  the 
length  of  the  unit  on  that  of  the  earth's  surface  was  to 
obtain  a  unit  which  would  be  permanent  and  reproducible 
without  reference  to  a  standard  if  the  latter  were  de- 
stroyed. A  more  logical  choice  of  unit  could  be  made 


SOME   USES  OF  MATHEMATICS  123 

to-day  as  a  result  of  our  greater  knowledge  of  physical 
magnitudes.  A  secondary  standard  was  suggested  by 
Michelson,1  the  recipient  in  1907  of  the  Nobel  prize. 
He  suggested  the  wave  length  of  light,  choosing  for  that 
purpose  the  red  light  emitted  by  cadmium  vapor  when 
an  electrical  discharge  passes  through  it.  In  terms 
of  this  wave  length  he  made  a  remarkably  accurate 
measurement  of  the  meter,  working,  of  course,  with  the 
prototype  at  Paris.  The  distance  represented  by  the 
meter  is,  therefore,  very  precisely  known  to-day  in 
terms  of  an  absolutely  stable  physical  magnitude 
which  may  be  reproduced  by  any  scientist  La  his  labora- 
tory. 

The  remaining  fundamental  units  from  which  the 
scientist  derives  all  the  other  units  are  those  of  mass 
and  tune.  The  unit  for  mass,  or  quantity  of  ponder- 
able matter,  is  the  gram  which  is  the  thousandth  part 
of  the  kilogram.  The  latter  was  originally  intended 
to  be  the  mass  of  a  cube  of  water,  one  tenth  of  a  meter 
on  a  side,  at  its  greatest  density.  Although  more 
recent  measurements  have  shown  that  it  is  not  accu- 
rately this  mass,  the  kilogram  is  the  accepted  standard. 
The  unit  of  time,  the  second,  has  been  discussed  in 
Chapter  III.  The  second,  the  hundredth  part  of  the 
meter  (1  cm.),  and  the  thousandth  part  of  the  kilo- 
gram (1  g.)  are  the  fundamental  units  for  scientific 
purposes. 

In  terms  of  these  fundamental  units  the  other  units 
are  easily  obtained  when  once  the  laws  are  known.  Such 
units  are  said  to  be  derived  units  of  the  C.  G.  S.  system. 

1  Cf. Michelson,  "Light  Waves  and  their  Uses  "  Univ.  of  Chicago 
Press,  1907. 


124         THE   REALITIES   OF   MODERN   SCIENCE 

Some  of  them  are  obtained  by  relations  of  the  general 
form  Z  =  XY,  which  are  not  physical  laws  but  defining 
equations.  For  example,  the  velocity  with  which  a 
body  moves  is  the  ratio  of  the  space  traversed  to  the 
time  consumed.  Hence  the  defining  equation  is 
V  =  S/T.  The  unit  of  velocity  is  then  1  cm./l  sec., 
or  1  cm.  per  sec.  as  it  is  usually  read. 

Similarly  if  the  velocity  changes,  that  is;  if  the  motion 
is  accelerated,  the  measure  of  the  acceleration  is  de- 
fined as  the  rate  of  change  of  velocity,  and  is  a  change 
in  velocity  (expressed  in  cm.  per  sec.)  occurring  in  1 
second.  Of  this  we  shall  have  occasion  to  treat  more 
fully  in  the  next  chapter  when  we  consider  "rates." 

Before  doing  so  it  may  be  of  interest  to  note  two 
illustrations  of  the  importance  of  mathematics  to  the 
student  of  science.  In  the  development  of  our  knowl- 
edge of  mechanics  and  hence  of  astronomy,  which  deals 
with  celestial  mechanics,  there  came  a  time  about 
1600  A.D.  when  further  progress  had  to  wait  until 
new  mathematical  tools  were  developed.  These  tools 
were  supplied  by  Newton,  who  invented  a  mathematical 
method  of  studying  problems  which  involve  motion. 
This  is  the  method  mentioned  on  page  32.  Similar 
methods  were  developed  independently  by  Leibnitz,  for 
this  was  a  remarkable  age  in  mathematics. 

The  second  illustration  concerns  the  discovery  of 
wireless  telegraphy.  In  1873  Maxwell,  who  was  a 
prominent  physicist,  highly  trained  in  the  use  of  mathe- 
matical tools,  announced  that  light  was  an  electrical 
phenomenon  and  traveled  as  an  electromagnetic  wave. 
He  further  stated  the  possibility  of  there  being  other 
electromagnetic  waves  which  would  not  produce  the 


SOME   USES  OF  MATHEMATICS  125 

effect  of  light  but  would  travel  just  as  light  waves 
travel. 

In  1887  Hertz  verified  this  prophecy  of  Maxwell 
and  announced  the  discovery  of  electromagnetic  waves. 
Hertz  studied  their  properties  or  characteristics.  He 
showed  how  they  could  be  produced,  how  they  traveled 
through  the  walls  of  buildings  and  were  not  affected 
by  obstacles  which  would  completely  obstruct  the  pas- 
sage of  light,  and  also  how  they  could  be  detected, 
since  they  do  not  affect  the  eyes  as  does  light. 

In  1896  Marconi  showed  how  these  waves  could  be 
utilized  for  telegraphy,  by  inventing  an  antenna  from 
which  they  might  start  out  and  by  which  they  might 
be  received. 

It  may  look  like  a  very  long  time  between  the  three 
steps  taken  by  Maxwell,  by  Hertz,  and  by  Marconi. 
But  one  must  remember  that  the  problem  which  con- 
fronted Hertz  was  that  of  producing  some  waves  in 
space  which  would  travel  like  light  although  they  were 
not  light  and  so  could  not  be  detected  by  the  human 
eye.  Some  instrument  had  to  be  devised  which  would 
act  toward  these  waves  just  as  the  eye  does  toward 
light  waves,  that  is,  which  would  indicate  their  presence. 
And,  after  what  one  thought  would  serve  for  an  "eye" 
had  been  made,  how  could  one  tell,  if  it  did  not  work, 
whether  the  fault  was  with  the  eye  or  with  the  apparatus 
which  was  expected  to  produce  the  waves? 

It  is  interesting  to  note  that  the  three  steps  were 
taken  by  men  whose  interests  and  abilities  in  physics 
were  perhaps  not  so  much  different  hi  amount  as  in 
kind.  Maxwell  was  interested  in  developing  a  com- 
plete theory,  which  naturally  took  a  mathematical 


126         THE   REALITIES   OF    MODERN   SCIENCE 

form,  for  explaining  electricity  and  magnetism.  He 
wanted  to  know  the  general  principles  and  laws  and  to 
put  them  into  such  form  that  men  might  predict  what 
would  happen  under  any  set  of  conditions  which  they 
might  imagine.  That  he  succeeded  very  well  is  evident 
from  his  contribution  to  radiotelegraphy.  Maxwell 
was  a  "  mathematical  physicist. " 

Hertz  although  well  trained  in  mathematics  was  a 
"  research  physicist. "  He  was  perhaps  most  interested 
in  experimental  attempts  to  extend  the  body  of  scien- 
tific knowledge.  He  was  not  immediately  concerned 
with  the  application  of  this  knowledge  to  the  uses  of 
mankind.  He  probably  knew  that  all  knowledge  is 
ultimately  of  use  and  that  his  contributions  to  the  sum 
total  of  general  knowledge  would  interest,  inform,  and 
inspire  others,  some  of  whom  might  make  practical 
applications. 

Marconi  was  the  inventor  and  the  engineer.  With- 
out his  vision  of  the  possibilities  of  Hertz's  discovery 
this  new  art  would  not  have  developed  until  some  time 
later.  He  contributed  some  of  the  means  for  the  prac- 
tical application. 

We  notice  that  all  three  types  were  necessary  for 
the  complete  development.  Sometimes  the  mental 
qualifications  are  combined  in  one  man,  but  in  most 
cases  a  man's  best  ability  lies  along  only  one  of  these 
lines.  If  we  see  the  importance  in  the  progress  of 
science  of  each  type,  we  may  appraise  more  accurately 
their  contributions  and  thus  avoid  the  popular  error 
of  attaching  too  much  credit  to  the  inventor  or  the 
converse  academic  error  of  failing  to  give  him  sufficient 
credit. 


SOME   USES  OF  MATHEMATICS  127 

The  illustration  in  question  indicates  well  the  manner 
by  which  science  grows,  that  is,  by  accretions  or  con- 
tributions. These  have  been  made  in  the  history  of 
science  by  many  men  whose  names  have  long  been 
forgotten  or,  as  hi  the  case  of  some  of  the  earliest  dis- 
coverers, by  men  who  probably  had  no  names  at  all. 
Such  contributions  have  been  made  by  men  of  all 
races  and  nationalities.  In  the  case  of  radio-telegraphy 
it  is  to  be  recalled  that  Maxwell  was  an  Englishman, 
Hertz  a  German,  and  Marconi  an  Italian. 

While  the  individual  who  makes  any  advance  in 
science  is  deserving  of  great  credit  it  must  not  be  for- 
gotten that  such  discoveries  are  rarely  if  ever  made  until 
the  time  is  ripe  for  them,  that  is,  until  the  whole  body 
of  scientific  knowledge  and  methods  has  prepared  the 
way  for  the  discovery  of  genius  or  of  accident,  as  the 
case  may  be.  The  man  who  actually  makes  the  dis- 
covery is  usually  but  a  little  time  ahead  of  his  foremost 
contemporaries. 

In  our  illustration  this  is  true  and  was  so  recog- 
nized, for  example,  by  Hertz.  In  the  preface  to  the 
collected  papers  on  his  work,  which  were  published  some 
years  later,  he  generously  but  truly  states  that  if  he 
had  not  happened  to  make  the  discovery,  some  other 
scientist  would  have  done  so  shortly.  He  mentions 
Lodge,  the  English  scientist,  who  was  then  working 
along  somewhat  similar  lines,  as  most  likely  to  have 
made  the  advance. 


CHAPTER  XI 

RATES 

WHEN  one  speaks  of  the  speed  of  a  passing  automobile 
as  so  many  miles,  say  thirty,  per  hour,  he  is  expressing 
his  estimate  of  an  instantaneous  rate,  for  he  does  not 
mean  that  the  machine  will  travel  thirty  miles  in  the 
hour  but  that  at  the  given  instant  its  motion  is  such 
that  a  continuance  at  this  rate  would  result  in  this 
displacement.  The  estimate  is  the  result  of  previous 
experiences,  conscious  or  otherwise,  in  estimating 
distances  and  times.  Although  the  observer  does 
not  make  actual  measurements  of  these  two  magni- 
tudes and  then  perform  an  arithmetical  operation 
upon  them,  the  method  is  fundamentally  that  which 
was  applied  in  the  "  speed  traps "  of  the  earlier  days 
of  automobiles.  In  such  traps  two  observation  stations 
were  established  along  the  roadway  at  measured  dis- 
tances apart  and  telephone  connections  were  provided 
between  the  observers.  The  time  consumed  by  any 
machine  in  traversing  the  known  distance  could  then 
be  obtained  and  hence  the  speed  in  miles  per  hour,  by 
taking  the  ratio  of  the  distance  in  miles  to  the  time  in 
hours. 

The  method  admitted  of  a  determination  only  of  the 
average  speed  through  the  trapped  distance.  Stories 
were  told,  therefore,  of  drivers  who  astonished  the 

128 


RATES  129 

observers  by  speeding  past  their  posts,  but,  because 
the  machines  were  slowed  down  in  the  intervening 
distance,  the  actual  figures  indicated  compliance  with 
the  law.  The  average  speed  was  only  necessarily  the 
actual  instantaneous  speed  at  each  observation  post  in 
case  the  car  traversed  the  entire  distance  at  a  uniform 
rate.  By  shortening  the  distance  for  which  the  car  is 
timed  a  result  more  nearly  approximating  the  actual 
speed  at  either  post  is  obtained,  but  this  is  because 
in  the  correspondingly  reduced  time  only  a  negligible 
change  in  speed  can  occur,  and  hence  the  observer  is 
dealing  with  an  essentially  uniform  speed  throughout 
the  measured  distance. 

The  general  method  which  is  indicated  for  obtain- 
ing the  actual,  that  is,  instantaneous,  speed  at  any 
point  on  the  path  of  a  moving  body  is  as  follows : 
Establish  an  observation  station  at  the  point  in  ques- 
tion and  a  second  station  as  near  as  possible1  to  the 
first.  Observe  the  time  required  for  the  body  to 
traverse  this  distance  and  find  the  ratio  of  the  distance 
to  the  time. 

We  recognize  that  this  method  requires  that  we  shall 
go  the  limit  in  reducing  the  separation  of  the  two 
stations.  Now,  our  observations  are  separated  not 
only  in  space  but  also  in  time,  and  it  is  usual  to  empha- 

1  Of  course,  in  practice  other  sources  of  error  than  the  change  in 
velocity  must  be  taken  into  account.  The  first  source  of  error  occurs 
in  the  observation  of  the  coincidences  of  the  moving  body  with 
the  two  reference  points,  and  the  second  is  found  in  the  measure- 
ments of  the  distance  and  the  corresponding  time  interval.  What- 
ever means  we  may  employ  we  only  approximate  the  magnitude, 
and  hence  in  exactness  we  should  always  express  each  measured 
magnitude  as  a  definite  amount,  plus  or  minus  a  limiting  error. 

K 


130         THE   REALITIES  OF   MODERN  SCIENCE 

size  the  latter  separation  by  saying  that  we  measure  the 
distance  traversed  between  two  points  in  time.  The 
ratio  of  the  distance  to  the  time  interval  is  the  average 
velocity.  As  the  time  interval  is  reduced  this  ratio 
becomes  more  and  more  nearly  representative  of  the 
instantaneous  rate.  Its  value,  in  other  words,  ap- 
proaches the  value  of  the  instantaneous  rate  as  a  limit, 
while  the  time  interval  is  caused  to  approach  zero  as 
a  limit. 

As  the  time  interval  between  observations  is  reduced 
the  distance  traversed  is  also  reduced.  The  fact  that 
both  the  numerator  and  the  denominator  of  the  ratio 
are  thus  made  "  infinitesimals "  does  not  mean  that  the 
ratio  may  not  have  a  perfectly  definite  and  finite  value. 
Of  this  idea,  that  a  ratio  of  two  magnitudes  may  be 
finite  even  though  the  magnitudes  concerned  are  physi- 
cally very  small,  we  have  previously  had  an  illustration 
in  our  comparison  of  the  distance  ratios  in  the  solar 
system  to  the  distance  ratios  of  the  electronic  systems 
of  molecules.  Physically  speaking,  with  reference  to 
the  diameter  of  the  earth  the  diameter  of  an  electron  is 
infinitesimal.  To  a  mathematician,  however,  an  in- 
finitesimal is  an  abstraction,  representing  a  quantity 
indefinitely  small,  which  approaches  zero  as  its  limiting 
value. 

The  fundamental  ideas  involved  in  the  concept  of  a 
rate  are  conveniently  illustrated  also  in  the  case  of  the 
slope  or  grade  of  a  road.  Such  a  slope  is  usually 
expressed  as  a  ratio  of  the  vertical  distance  to  the 
corresponding  horizontal  distance,  e.g.  as  a  two  per 
cent  grade.  It  is  obviously  the  rate  at  which  one 
ascends  with  respect  to  his  horizontal  displacement. 


RATES 


131 


0 


FIG.  9. 


H 


The  analytical  processes,  involved  in  the  study  of  rates, 
follow  from  a  consideration  of  this  case. 

In  Fig.  9  is  given  the  cross  section  of  a  road.  Axes, 
OH  and  OF,  permit  the  expression  of  the  location  of 
any  point  of  the 
road  with  reference  v 
to  their  intersection. 
For  convenience  we 
shall  denote  dis- 
tances measured 
along  OH  and  OF 
by  H  and  V,  respec- 
tively. Correspond- 
ing to  any  chosen 

value  of  H ,  or,  as  we  may  say,  to  any  value  which  H 
may  assume,  there  is  a  value  of  F  which  satisfies  the 
condition  that  the  point  determined  by  these  "  coor- 
dinates" lies  on  the  curve  representing  the  road.  In 
other  words,  as  H  varies,  F  also 
varies,  being  dependent  for  its  value 
upon  H.  It  is  usual,  therefore,  to 
say  that  H  is  an  independent  vari- 
able and  that  F,  the  dependent 
variable,  is  a  function  of  H.  In 
the  particular  case  shown  in  the 
figure  it  is  evident  that  the  slope 
varies  from  point  to  point,  that  is, 
that  the  slope  itself  is  a  function 
of  H. 

Between  any  two  points,  as  1  and  2,  represented  in 
the  enlargement  of  Fig.  10  by  the  coordinates  HI,  Fi, 
and  H2,  F2,  respectively,  the  average  slope  is  the 


', 


•  • 


FIG.  10. 


132         THE   REALITIES  OF   MODERN  SCIENCE 

quotient  of  V2—Vi  and  H2—Hi.  This  is  evidently 
the  slope  of  the  dotted  line  extending  from  1  to  2.  As 
the  point  2  is  chosen  closer  to  1  (that  is,  as  the  incre- 
ment, H2— HI,  approaches  zero),  the  dotted  line  be- 
comes more  nearly  tangent  to  the  curve  at  the  point 
1.  The  actual  rate  of  change  of  V  with  respect  to  H, 
therefore,  at  any  given  point  is  the  average  slope  of  a 
tangent  to  the  curve  at  this  point.  We  may  find  the 
rate  graphically,  therefore,  as  accurately  as  we  can 
construct  the  tangent.1  In  general,  we  may  define 
a  rate  as  the  limit  approached  by  the  ratio  between 
corresponding  increments,  in  the  dependent  and  in- 
dependent variables  re- 
spectively, as  the  in- 
crement of  the  latter 

approaches  zero. 

In  this  particular  case 
of  the  slope  of  a  road 
the  variables  are  both 
t     distances,     which     are 


o  ti  tz       ta     t4  measured   and  plotted 

FlG-  n-  with  reference  to  a  com- 

mon reference  point.  A  similar  plot  may  be  made  for 
a  case  involving  unlike  variables,  e.g.  that  of  the  moving 
automobile.  For  this  we  must  assume  reference  points 
both  in  time  and  in  space.  Corresponding  to  any  time, 
say  t,  subsequent  to  our  assumed  zero  of  time,  the  auto- 

1  Analytical  methods  of  absolute  accuracy  are  possible  if  the 
form  of  the  function,  which  one  variable  is  of  the  other,  may  be 
expressed  in  symbols.  The  methods  are  those  of  the  calculus. 
For  example,  in  a  function  like  s  =  at2/2  the  rate  of  change  of  s 
with  respect  to  t  (usually  symbolized  as  ds/dt)  may  be  shown  to 
be  at. 


RATES  133 

mobile  would  be  separated  by  a  distance  s  from  the 
reference  point  in  space.  A  plot  may,  therefore,  be  con- 
structed as  in  Fig.  11.  The  rate  of  change  of  position 
with  respect  to  a  change  in  time,  that  is,  the  speed  of 
the  car,  is  given  at  each  instant  of  time  by  the  slope  of 
a  tangent  to  the  corresponding  point  of  the  curve.  It 
is  therefore  evident  that  for  the  concrete  case  of  the 
figure  the  car  is  originally  at  rest  and  so  remains  until 
a  time  ti  has  elapsed.  At  a  time  ^  the  maximum 
speed  is  attained.  This  speed  is  maintained  until 
£3,  following  which  the  car  is  brought  gradually  to 
rest  at  t4.  It  remains  at  rest  for  the  balance  of  the 
time  indicated  by  the  plot. 

This  slope,  involving  unlike  magnitudes,  may  be 
termed  a  physical  rate  as  distinct  from  the  geometrical 
rate,  illustrated  by  the  slope  of  a  road.  The  only 
difference  is  that  of  units,  for  geometrical  rates  are 
pure  numbers  but  a  physical  rate  is  expressed  by  a 
numeric  and  a  compound  unit.  Thus  the  unit  of  speed 
is  the  quotient  of  unit  length  and  unit  time,  e.g. 
1  cm./l  sec. 

The  most  familiar  rates  of  everyday  life  are  those 
of  speed  and  interest,  both  time-rates,  if  we  name 
them  for  their  independent  variables.  For  the  scien- 
tist, of  course,  there  are  as  many  different  kinds  of 
rates  as  there  are  physical  magnitudes  which  may  be 
considered  independent  variables.  For  example,  he 
may  be  concerned  with  a  rate  of  expansion  with  respect 
to  temperature,  with  a  rate  of  increase  of  electric 
current  with  electromotive  force,  or  with  a  rate  of 
change  of  energy  with  respect  to  space.  In  the  following 
chapter  we  shall  see  how  force  is  a  space  rate  of  change 


134         THE   REALITIES  OF   MODERN  SCIENCE 

of  energy.  For  that  discussion  we  shall  need  a  relation- 
ship between  the  velocity  acquired  by  a  body  which 
is  uniformly  accelerated  and  the  space  over  which  it 
passes. 

The  scientist  uses  the  word  " velocity"  to  represent 
speed  in  a  definite  direction.  For  example,  a  point 
on  a  uniformly  rotating  wheel  is  moving  with  constant 
speed,  but  its  velocity  is  constantly  changing,  for  the 
direction  of  motion  is  constantly  changing.  If  we  are 
dealing,  however,  only  with  motion  in  a  straight  line, 
velocity  and  speed  are  identical.  Just  as  velocity  is 
defined  as  the  time-rate  of  change  of  position,  so 
acceleration  is  defined  as  the  time-rate  of  change  of 
velocity.  (In  the  illustration  of  the  rotating  wheel 
it  is  evident  that  each  point  is  constantly  accelerated 
toward  the  center.) 

The  simplest  case  for  analysis  is  that  of  a  body 
moving  with  constant  velocity.  If  the  velocity  is 
v  it  will  travel  in  a  time,  t,  a  distance,  s,  such  that 
s  =  vt.  If  the  velocity  is  not  constant  the  simplest 
case  occurs  when  the  acceleration  is  uniform,  that  is, 
when  the  velocity  increases  the  same  number  of  units 
of  velocity  in  each  unit  of  time.  Let  a  represent  the 
acceleration,  then  if  the  initial  velocity  is  zero  the 
velocity  at  the  end  of  t  seconds  is  expressed  as  v=at. 
(In  general,1  if  the  initial  velocity  is  not  zero  but  is  00, 
then  the  velocity  at  the  end  of  t  seconds  is  v  =  v0+at.) 

Because  the  velocity  increases  uniformly  it  follows 
that  the  body  is  moving  with  its  average  velocity  at 

1  In  an  ordinary  elementary  course  in  physics  there  occur  about 
twenty  physical  relations  which  are  entirely  analogous  to  this 
expression  and  are  represented  by  similar  equations. 


RATES  135 

the  middle  of  the  interval,  t,  and  also  that  for  each 
instantaneous  velocity  below  this  average  there  is  a 
corresponding  velocity  equally  above  the  average. 
So  far  as  concerns  the  total  distance  traversed  in  the 
time  t  the  result  is  the  same  as  if  the  body  moved  with 
a  uniform  velocity,  v,  equal  to  the  average  value.  Con- 
sider then  the  case  of  a  body  uniformly  accelerated 
from  rest.  The  average  velocity  is  half  the  sum  of  the 
initial  and  final  velocities,  that  is  v  =  at/2.  The  dis- 
tance traversed  is  s  =  vt  and  hence  s  =  at2/2.  (For 
example,  in  the  case  of  a  body  falling  freely  from  rest 
the  acceleration  is  of  value  "g"  and  s  =  gt2/2.)  The 
distance  is  therefore  proportional  to  the  square  of  the 
time. 

For  our  later  purposes,  however,  we  wish  to  express 
the  relation  between  the  total  space  traversed  and  the 
final  velocity  which  the  body  has  acquired.  This 
velocity  is  v  =  at  and  hence  t  =  v/a.  Substituting 
v2/a2  for  t2  in  s  =  at2/2  gives  s  =  v2/2a  or  v2=2as,  as 
the  desired  relation.  The  distance,  therefore,  in- 
creases as  the  square  of  the  velocity. 

The  three  relations  for  free  fall  from  rest,  namely, 
v  =  gt,  s  =  gt2/2  and  v2=2gs  were  derived  by  Galileo, 
whose  experiments  from  the  leaning  tower  of  Pisa 
were  more  spectacular  but  less  fruitful  in  scientific 
development  than  those  of  his  other  methods.  His 
study  followed  the  line  of  assumption  and  experimen- 
tation. Several  assumptions  he  disproved  himself  by 
further  analysis  before  he  hit  upon  the  correct  one,  for 
in  his  day  acceleration  was  unknown  and  its  concept 
is  due  to  him.  Starting,  finally,  with  the  assumption 
that  if  two  bodies  are  allowed  to  fall  from  rest,  one, 


136         THE  REALITIES  OF   MODERN  SCIENCE 

falling  for  twice  the  time  of  the  other,  will  acquire  twice 
the  velocity,  he  formulated  the  necessary  relations  for 
s  and  t.  He  had  to  deal  with  s  and  t  instead  of  v  and  t 
because  he  had  no  means  for  measuring  instantaneous 
velocities.  His  reasoning  has  been  followed  in  the 
preceding  analysis,  where  it  was  shown  that  the 
total  distance  traversed  should  be  proportional  to 
the  square  of  the  time  during  which  motion  occurs. 

This  conclusion  from  his  fundamental  assumption 
he  set  himself  to  test.  Clocks  and  chronographs  were, 
of  course,  unknown  and  only  water  clocks  or  sand- 
glasses were  available.  With  such  crude  devices  it 
would  ordinarily  be  impossible  to  determine  the  time 
of  descent  of  a  freely  falling  body  with  sufficient 
accuracy  to  check  the  conclusion.  He  sought  first, 
therefore,  to  retard  the  descent  so  that  it  might  be 
more  accurately  observed.  For  this  reason  he  studied 
the  motion  of  balls  rolling  down  grooves  in  an  in- 
clined plane.  A  further  assumption  was  therefore 
required  to  the  effect  that  the  form  of  the  relation- 
ship between  s  and  t  would  not  be  altered.  Measuring 
distances  from  the  upper  extremity  of  the  plane  he 
marked  off  lengths  of  1,  4,  9,  and  16  arbitrary  units. 
The  corresponding  times  of  descent  should  be  found 
to  be  as  1,  2,  3,  and  4. 

He  next  sought  an  improvement  in  methods  of 
timing.  For  the  means  he  used  a  vessel  of  water, 
with  a  large  transverse  area,  in  the  bottom  of  which 
there  was  a  minute  orifice  which  he  could  close  by  his 
finger.  Such  a  vessel  will  have  a  practically  constant 
" pressure  head"  while  a  small  amount  of  water  is 
being  discharged,  and  hence  the  weights  of  water 


RATES  137 

discharged  will  be  proportional  to  the  time  of  flow.1 
With  this  apparatus  he  confirmed  his  assumptions. 

Further  studies  by  Galileo  of  the  kinematics  of 
uniformly  accelerated  bodies  contributed  much  to  that 
base  of  accumulated  knowledge  upon  which  his  suc- 
cessors, and  especially  Newton,  built  the  classical 
system  of  mechanics.  The  modern  concept  of  energy 
upon  which  science  rests  to-day  was  frequently  ap- 
proached but  apparently  never  firmly  grasped  by  these 
natural  philosophers.  For  example,  Galileo  studied 
the  relations  which  must  subsist  between  free  descent 
and  motion  on  an  inclined  plane.  As  he  had  shown 
previously,  the  velocity  of  a  freely  falling  body  is 
increased  proportionally  to  the  time  of  descent.  He 
therefore  reasoned  that  if  its  direction  of  motion  should 
be  reversed  at  any  instant,  as  by  a  reflection,  the 
velocity  would  then  be  diminished  proportionally 
to  the  time  of  ascent.  The  body  should,  therefore, 
rise  through  the  same  distance  as  it  had  previously 
fallen.  If,  then,  a  body  is  allowed  to  roll  down  an 
inclined  plane  and  up  another  it  cannot  rise  to  a  point 
higher  than  that  from  which  it  started.  In  the  limiting 
case  it  will  just  attain  this  height.  If  it  could  exceed 
this  height  it  would  be  possible  to  arrange  inclined 
planes  so  as  to  effect  the  elevation  of  bodies  by  gravity 
alone.  The  velocity  acquired  in  frictionless  descent 
along  an  inclined  plane  depends,  then,  only  upon  the 

1  It  was  Torricelli,  somewhat  later,  who  perceived  the  analogy 
between  freely  falling  bodies  and  the  flow  of  liquids  under  a  gravity 
feed.  He  showed  that,  neglecting  all  frictional  resistances,  the 
velocity  of  efflux  from  an  orifice,  a  distance  of  h  below  the  surface, 
is  given  by  v*  =  2gh.  Hence  if  the  head,  h,  is  constant  the  velocity 
is  constant  and  the  discharge  is  proportional  to  the  time. 


138         THE  REALITIES  OF   MODERN  SCIENCE 

vertical  height  and  not  upon  the  slope  of  the  plane. 
As  Mach1  points  out  there  is  contained  in  Galileo's 
assumption  "the  uncontradictory  apprehension  and 
recognition  of  the  fact  that  heavy  bodies  do  not  pos- 
sess the  tendency  to  rise  but  only  to  fall."  The  ideas 
involved,  however,  are  really  those  of  energy,  for 
obviously  the  energy  acquired  in  falling  must  be 
equal  in  a  conservative  system  to  the  gain  in  energy 
which  would  be  consequent  to  a  reversal  of  the  dis- 
placement. 

1  Mach,  "Science  of  Mechanics,"  pp.  134  et  seq. 


CHAPTER   XII 
FORCE,  A  SPACE  RATE  OF  ENERGY 

IN  our  earlier  discussion  we  have  used  the  word 
''force"  without  definition,  since  we  all  have  a  con- 
cept acquired  by  our  own  muscular  experiences.  The 
physicist,  however,  has  given  to  this  term  a  technical 
meaning.  Thus  as  one  of  the  definitions  in  the  Century 
Dictionary  there  is  "the  immediate  cause  of  a  change 
in  the  velocity  or  direction  of  motion  of  a  body." 
This  concept  originated  in  1647  with  Newton,  who 
stated  three  fundamental  laws  as  to  the  motion  of 
bodies,  which  he  had  deduced  from  the  facts  of  astron- 
omy and  from  various  experiments.  His  first  law  says 
that  "every  body  continues  in  a  state  of  rest  or  of 
uniform  motion  in  a  straight  line  except  in  so  far 
as  compelled  by  force  to  change  that  state."  This 
statement  defines  force,  and  implies  a  characteristic 
of  all  bodies,  inertia.  Bodies  are  inert. 

That  this  statement  of  Newton  is  really  a  law  we 
all  believe  although  we  cannot  prove  it.  We  know 
that  bodies  at  rest  remain  at  rest  unless  they  are 
forcibly  moved  and  that  it  requires  force  to  change 
the  direction  of  motion  of  a  body.  As  to  the  idea 
that  a  body  would  continue  in  uniform  motion  in  a 
straight  line  if  no  force  were  exerted  upon  it  we  have 
no  proof.  If  we  start  a  body  it  comes  to  rest,  sooner 

139 


140         THE   REALITIES  OF   MODERN  SCIENCE 

or  later,  because  of  the  "frictional  forces."  If  the 
friction  is  reduced  the  body  maintains  a  state  of 
motion  for  a  longer  time.  We,  therefore,  reason  that 
if  all  opposing  forces  could  be  removed  the  body  would 
continue  to  move  with  uniform  velocity. 

This  law  was  stated  about  two  centuries  before  the 
reality  of  energy  had  been  recognized.  Changes  in 
energy  are  the  concomitants  of  changes  in  motion  and 
to-day  many  scientists  consider  the  former  to  be  the 
causes  of  the  latter.  We  have  seen  that  energy  may 
be  either  potential  or  kinetic  and  also  that  decreases 
in  potential  energy  result  in  increases  in  kinetic  energy 
or  vice  versa. 

A  body  in  motion  or  one  at  rest  will  have  the  same 
kinetic  energy  indefinitely  unless  it  acquires  more  at 
the  expense  of  some  source  of  energy  or  unless  it  im- 
parts its  energy  to  other  bodies  or  to  parts  of  itself. 
The  latter  happens  when  there  is  friction,  for  the  body 
then  transfers  some  of  the  kinetic  energy,  which  it 
has  as  a  bulk,  to  its  own  and  adjacent  molecules. 

A  moving  body  is  retarded  if  its  motion  is  in  such  a 
direction  as  to  increase  the  potential  of  the  system  which 
it  forms  with  another  body.  As  a  body  which  is 
thrown  into  the  air  moves  away  from  the  earth  the 
potential  energy  of  the  system  increases.  The  kinetic 
energy  of  the  body  is  thus  converted  into  potential 
energy,  and  at  the  instant  when  the  conversion  is  just 
completed  the  body  is  at  rest.1  It  then  falls  and  the 

1  Of  course,  if  the  kinetic  energy  is  greater  than  the  greatest 
amount  of  potential  energy  which  the  body  can  add  to  the  system 
by  moving  to  any  distance  whatever,  then  it  cannot  thus  come  to 
rest.  This  is  the  case  for  comets. 


FORCE,   A   SPACE  RATE  OF  ENERGY  141 

potential  energy  is  in  turn  converted  into  kinetic 
energy. 

During  the  centuries,  however,  which  followed  New- 
ton and  before  the  modern  ideas  of  energy  were  ac- 
cepted, the  word  "force"  in  the  Newtonian  sense  be- 
came firmly  fixed  in  scientific  language.  Thus  scien- 
tists to-day  speak  of  frictional  forces  as  if  these  were 
the  causes  of  the  slowing  down  of  a  moving  body  even 
though  they  realize  that  changes  in  the  form  and  loca- 
tion of  its  energy  are  the  real  causes. 

In  the  case  of  the  gravitation  of  bodies  toward  the 
earth  we  still  speak  of  the  "force"  of  gravitation  in- 
stead of  saying  that  the  earth  and  any  other  body  form 
a  system,  the  potential  energy  of  which  decreases  as 
they  approach  one  another.  The  reader  will  find 
this  phenomenon  almost  always  described  in  terms  of 
force  as  a  cause.  In  part  this  is  due  to  the  form  in 
which  Newton  expressed  his  so-called  Law  of  Universal 
Gravitation.  This  states  that  any  two  bodies  (or 
strictly  "particles")  attract  each  other  with  a  force 
which  is  proportional  to  the  product  of  their  masses 
and  inversely  as  the  square  of  the  distance  between 
their  centers.1  If  Newton  had  lived  after  the  Prin- 
ciple of  the  Conservation  of  Energy  had  been  accepted 
it  is  quite  probable  that  he  would  have  expressed  his 
law  of  gravitation  in  terms  of  energy  instead  of  force. 
Later  we  shall  see  how  nearly  he  came  to  recognizing 

1  If  the  masses  are  mi  and  ra2  and  the  distance  between  centers 
is  r,  then  the  force,  /,  is 


or/  =  (1) 


142         THE   REALITIES  OF   MODERN  SCIENCE 

in  his  third  law  of  motion  the  idea  of  the  conserva- 
tion of  energy. 

The  Law  of  Universal  Gravitation  may  be  stated  as 
follows  :  Any  two  bodies  form  a  system  the  potential 
energy  of  which  tends  to  decrease  and  is  always  less 
than  it  would  be  if  the  bodies  were  infinitely  far  apart 
by  an  amount  proportional  to  the  product  of  the  masses 
and  inversely  as  the  distance  between  centers.  Thus 
let  Px  represent  the  potential  energy  when  the  bodies 
are  at  an  infinite  distance  from  each  other,  and  Pr  the 
potential  energy  when  they  are  separated  by  some 
finite  distance,  r.  Let  mi  and  m^  be  the  masses  of  the 

two  bodies,  then 

(2) 


where  k  is  a  factor  which  reduces  m*m2  to  the  same  units 

r 

as  are  used  in  expressing  Pr  and  P^. 

As  a  special  case  consider  a  body  of  mass  m  resting 
on  the  surface  of  the  earth.  Represent  the  mass  and 
radius  of  the  earth  by  M  and  R  respectively.  Then 
the  potential  energy  of  the  system  is 

(3) 

If  the  mass  m  is  raised  a  distance  x  above  the  surface 
the  distance  r  of  equation  (2)  is  no  longer  R  but  is 
R+x,  and  hence 

P      -p   -k  Mm  (4\ 

^R+x 

The  increase  in  potential  energy  occasioned  by  raising 
the  body  is  found  by  subtracting  (3)  from  (4)  thus 


FORCE,  A  SPACE  RATE  OF  ENERGY  143 

)       (5) 


P       _p        kMm     kMm  _  kMm  f     x 
R        R+x~      R     \R+x 


Now  R  is  about  3960  miles.  If  we  are  concerned  only 
with  separations  which  are  small  as  compared  to  3960 
miles  we  have  R+x  approximately  equal  to  R  and 
hence  we  may  obtain  from  equation  (5)  an  approxi- 
mate l  value. 

p      _p  _kMmx  xgx 

It  follows  that  near  the  surface  of  the  earth  the  in- 
crease in  potential  energy  is  directly  proportional  to 
the  increase  in  separation.  Dividing  this  increase  by 
the  distance  x  we  obtain  as  the  rate  2  at  which  the 
p.e.  increases  with  distance,  kMm/R2.  In  other  words, 
the  p.e.  of  the  system  formed  by  the  earth  and  a  mass 
m  increases  kMm/R2  energy  units  for  each  unit  of 
length  by  which  the  separation  increases. 

Now  the  earth  is  an  oblate  sphere  with  its  larger 
radius  at  the  equator.  Raising  a  given  mass  a  given 
distance  would  produce  a  smaller  increase  in  p.e. 
at  the  equator  than  at  the  poles  since  R  would  be 
greater  at  the  equator.  In  general  the  change  in  p.e. 
corresponding  to  the  vertical  movement  of  one  gram 
through  one  centimeter,  will  depend  upon  the  region 
of  the  earth's  surface  where  this  displacement  takes 
place.  The  gravitational  unit  of  energy,  the  gram 

1  For  example  if  x  is  1  mile  the  increase  in  potential  energy  as 
given  by  equation  (6)  is  about  one  quarter  of  a  per  cent  larger  than 
as  given  by  (5). 

2  The  space  rate  of  change  of  p.e.  is  then  of  the  form  •   n^lt 

which  is  the  force  according  to  Newton's  definition.  Cf.  foot- 
note p.  141. 


144        THE  REALITIES  OF   MODERN  SCIENCE 

centimeter  which  we  used  in  Chapter  IX,  is  conse- 
quently not  invariable.  We  need  an  absolute  unit, 
and  we  shall  now  derive  it  from  the  three  fundamental 
units  of  length,  mass  and  time. 

Since  the  p.e.  which  is  available  for  conversion  into 
kinetic  energy  when  1  gram  falls  1  cm.  depends  upon 
the  locality,  the  k.e.  which  is  acquired  is  similarly 
dependent.  If  a  gram  in  falling  1  cm.  acquires  a 
greater  k.e.  in  one  locality  than  in  another  it  must 
also  have  acquired  a  greater  velocity.  That  is,  it  is 
accelerated  more  and  the  value  of  g  should  be  higher. 
Let  us  therefore  measure  energy  in  such  a  unit  that  the 
numeric  expressing  1  gram-cm,  shall  be  proportional 
to  the  acceleration  at  the  given  locality.  We  have 
already  seen  that  the  increase  in  energy  is  proportional 
to  the  total  distance  traversed  and  to  the  mass  of  the 
displaced  body.  Hence,  let  us  write  as  the  defining 
equation  of  the  unit  of  energy 


W8-W0  =  mas  (7) 

when  W0  is  the  initial  energy  of  a  body  of  mass  m 
which,  after  traveling  a  distance  s  with  a  uniform 
acceleration  of  a,  has  a  final  energy  Ws. 

We  now  apply  the  method  of  Chapter  X  by  placing 
m  =  1  gram,  s  =  1  cm.,  and  a  =  1cm./  1  sec.2,  then  Ws  —  W0 
is  one  unit  of  energy.  Unit  energy  is  that  expended 
upon  (and  hence  that  acquired  by)  1  gram  in  moving 
1  cm.  with  an  acceleration  of  1  cm.  per  sec.  per  sec. 
This  unit  is  so  frequently  used  that  it  has  been  given  a 
convenient  name,  viz.  the  "erg"  from  the  same  Greek 
root  as  "energy." 

If  a  in  equation  (7)  has  the  special  value  of  g  then 


FORCE,   A   SPACE  RATE  OF  ENERGY  145 

for  m  =  l  g.  and  s  =  l  cm.  the  change  in  energy  is  g 
ergs.     Since  g  is  about  980  cm.  /  sec.2  it  follows  that  1 
gram-centimeter  is  about  980  ergs.1 
Considering  equation  (7)  we  see  that 


(8) 


states  the  rate  at  which  the  energy  changes  as  the 
separation  s  is  altered.  This  space  2  rate  of  change  of 
energy  is  defined  as  the  force.  The  unit  of  energy 
being  the  erg,  the  unit  of  force  is  1  erg  per  cm.  This 
unit  is  called  the  "dyne"  from  the  root  which  we 
recognize  hi  our  word  "dynamic."  It  is  also  evident 
that  the  dyne  is  the  force  which  will  accelerate  1  g. 
unit  amount,  that  is,  change  the  velocity  unit  amount 
(1  cm.  per  sec.)  each  second. 
According  to  this  definition  forces  are  called  into 

1  A  pound  is  454  g.  and  a  foot  is  30.5  cm.     It  therefore  follows 
that  1  ft.  Ib.  of  work  is  (980)  (454)  (30.5)  or  13,600,000  ergs.     The 
erg  is  obviously  too  small  a  unit  for  practical  purposes.     A  multiple 
known  as  the  "joule"  and  equal  to  ten  million  (107)  ergs  is  there- 
fore used. 

2  Distinguish  between  the  space  rate  of  energy,  which  is  force, 
and  the  time  rate,  which  is  power. 

Frequently  we  are  interested  in  the  time  rate  at  which  energy 
is  expended.  The  unit  of  1  erg  per  second  is  inconveniently  small 
and  the  practical  unit  of  one  joule  per  second  is  known  as  the  watt. 
A  thousand  joules  per  second  (1010  ergs)  is  the  kilowatt,  which  is 
familiar  to  all  purchasers  of  electrical  energy.  In  the  3600  seconds 
of  an  hour  during  which  energy  is  received  at  the  rate  of  1  kilowatt 
there  is  received  a  total  of  3.6  million  joules  or  36  million  million  ergs. 
In  such  units  one's  household  consumption  of  electrical  energy  appears 
enormous.  It  might  be  noted,  however,  that  the  energy  released 
by  the  combustion  of  one  pound  of  coal  (and  the  necessary  air)  is 
about  13  million  joules  or  enough  for  almost  4  kw.  hrs.  if  there  were 
no  dissipation  in  the  transformation. 


146         THE   REALITIES  OF   MODERN   SCIENCE 

play  only  when  changes  occur  in  the  energy  of  a  system, 
either  in  its  total  amount  or  in  its  kind  or  location. 
From  this  viewpoint  a  table  does  no  work  in  keeping 
a  book  from  falling,  for  there  is  no  change  in  the  energy 
of  the  book  and  the  ground  and  hence  no  force  is 
exerted  by  the  table.  The  difficulty  in  which  this 
definition  of  force  involves  us  is  obvious.  The  trouble, 
however,  is  not  due  to  our  present  definition  but  to 
the  fact  that  our  scientific  language  lacks  as  yet  ade- 
quate terminology.  What  we  actually  do  is  to  use 
the  word  in  the  above  rigorous  sense  and  also  in  the 
less  technical  sense  of  our  earlier  chapters. 

We  recognize  that  if  the  supporting  table  is  with- 
drawn the  book  will  fall  and  a  force  will  be  exerted  in 
accelerating  it.  But  the  book  falls  not  because  of  a 
force  but  rather  because  of  the  partial  conversion  of 
its  potential  energy  into  kinetic  energy.  We  do  not 
as  yet  know  why  such  a  change  in  energy  occurs  but 
it  is  a  phenomenon  of  nature  that  it  does  unless  some 
obstacle  like  the  table  intervenes.  We  have  no  word 
to  express  this,  idea,  although  the  word  " tractate" 
has  been  proposed.1  Similarly  "pellate"  has  been 
suggested  to  describe  the  natural  motion  of  bodies 
away  from  each  other.  For  example,  like  electrical 
charges  pellate,  unlike  tractate. 

In  the  case  of  a  falling  body  mg  dynes  is  the  force 
exerted  when  it  falls  freely.  (This  follows  at  once  from 
equation  (8)  by  substituting  g  for  a.)  If  the  energy 
released  as  it  falls  is  not  available  for  acceleration 
then  the  space  rate  of  change  of  the  energy  available 

1  By  Frederick  Soddy  in  "Matter  and  Energy,"  Holt,  Home 
Univ.  Library,  pp.  110-111. 


FORCE,  A  SPACE  RATE  OF  ENERGY  147 

for  accelerating  the  body  will  be  less  than  mg  and  may 
even  be  zero,  as  was  the  case  in  the  experiment  of 
Joule  described  on  page  112.  The  value  mg  is  thus  the 
maximum  force  which  can  be  called  into  play  by  allow- 
ing the  potential  energy  to  decrease.  This  maximum 
force  which  a  body  can  exert  in  free  fall  is  called  its 
weight.  The  weight  of  1  gram  is  then  g  dynes,  or 
approximately  980  dynes,  depending  upon  the  locality. 

This  inexact  use  of  the  word  " force"  as  meaning 
"  weight,"  that  is,  the  maximum  space  rate  at  which 
energy  will  be  released  if  gravitational  tractation  is 
allowed,  persists  in  the  language  of  physics.  There 
is  no  serious  objection  to  this  use  provided  that  it 
does  not  encourage  an  attitude  of  considering  force 
as  a  cause  instead  of  a  rate.  Energy  is  the  cause  in 
terms  of  which  we  must  seek  to  explain  all  physical 
phenomena. 

The  rigorous  quantitative  meaning  which  modern 
physics  has  given  to  " force"  leads  to  the  same  numeri- 
cal measure  as  Newton  selected  and  stated  in  his 
second  law.  This  law  says  that  "rate  of  change  of 
quantity  of  motion  is  proportional  to  the  force  and  takes 
place  in  the  straight  line  in  which  the  force  acts."  To- 
day "quantity  of  motion"  is  called  "momentum."  It 
is  measured  by  the  product  of  mass  and  velocity.  Since 
only  the  velocity  changes  and  since  the  rate  of  change 
of  velocity  is  acceleration,  the  rate  of  change  of  momen- 
tum is  the  product  of  mass  and  acceleration.  From 
equation  (8)  it  is  evident  that  the  two  definitions  are 
equivalent,  for  the  left-hand  member  expresses  our 
present  definition  and  the  right-hand  the  Newtonian 
definition. 


148         THE   REALITIES  OF    MODERN  SCIENCE 

Newton  recognized  that  the  change  in  the  momen- 
tum of  a  body,  which  takes  place  when  it  is  acted  upon 
by  a  force,  is  merely  one  aspect  of  the  phenomenon. 
He  reached  a  conclusion,  partly  by  experiment  and 
partly  by  inference,  that  in  all  such  cases  an  equal 
and  opposite  change  in  momentum  must  occur  in  some 
other  body.1  As  he  stated  it  in  his  third  law  of  motion: 
" Action  and  reaction  are  equal  and  opposite;  or  in 
other  words,  the  mutual  actions  of  two  bodies  are 
always  equal  and  oppositely  directed."  We  recog- 
nize this  fact  in  such  cases  as  that  of  a  man  jumping 
from  a.  small  boat.  As  the  man  jumps  forward,  he 
kicks  the  boat  backward.  The  man  and  the  boat  are 
put  into  motion  in  opposite  directions.  According  to 
Newton,  the  momentum  of  the  man,  that  is,  the  product 
of  his  mass  and  his  velocity,  is  equal  to  the  momentum 
of  the  boat.  If,  then,  the  boat  has  a  large  mass  it  will 
acquire  only  a  small  velocity.  If  its  mass  is  very  large 

JIt  is  therefore  usual  to  say  that  there  is  a  "conservation"  of 
momentum.  The  use  of  the  term  is  unfortunate,  particularly  in 
elementary  textbooks,  for  it  implies  that  momentum  is  an  indestruct- 
ible in  the  same  sense  as  is  energy.  To  readers  who  remember 
the  emphasis  in  such  texts  it  may  have  appeared  that  momentum 
is  a  reality  like  energy.  The  principle  of  the  conservation  of  mo- 
mentum is  a  recognition  of  the  characteristic  inertia  of  matter  which 
is  implied  in  Newton's  first  law  of  motion.  The  parts  of  a  system 
cannot  of  themselves  alter  the  location  of  their  center  of  mass.  For 
example,  at  any  instant  after  the  explosion  of  a  shell  the  configura- 
tion of  the  fragments  will  be  such  that  their  center  of  mass  occupies 
the  position  in  space  which  the  center  of  mass  of  the  shell  would  have 
occupied  at  that  instant  if  the  explosion  had  not  occurred.  The 
principle  is  equivalent  to  that  of  the  "conservation  of  the  center 
of  mass."  Both  are  implicit  consequences  of  inertia,  the  distinctive 
quality  of  bodies  which  may  acquire  kinetic  energy.  Momentum 
is  not,  therefore,  to  be  considered  a  reality  in  the  sense  in  which 
the  word  has  been  applied  to  matter  and  to  energy. 


FORCE,  A   SPACE  RATE  OF  ENERGY  149 

as  compared  to  that  of  the  man  this  velocity  may  well 
be  both  imperceptible  and  negligible. 

That  actions  occur  in  pairs,  equal  and  opposite,  we 
recognize  in  our  daily  acts.  Thus  if  one  wishes  to  exert 
a  force  he  braces  himself  against  some  firm  backing, 
that  is,  something  of  such  mass  that  the  momentum  he 
imparts  to  it  will  produce  but  a  negligible  velocity. 
The  firmest  backing  is,  of  course,  the  earth.  Its  mass 
is  so  large  that  any  motion  which  we  can  impart  to  it, 
as  for  example  by  shooting  a  shell  from  a  howitzer,  is 
so  small  that  we  frequently  overlook  the  fact  that  the 
action  is  similar  to  that  of  the  man  jumping  from  the 
boat. 

Consider  the  stresses  in  the  case  of  a  man  lifting  a 
mass  from  the  earth.  As  he  pushes  up  he  also  pushes 
down  on  the  earth.  There  are  then 
two  pairs  of  mutual  actions,  namely 
those  between  the  man  and  the  mass 
and  those  between  the  man  and  the 
earth.  The  man  pushes  up  on  the 
body,  its  reaction  is  down  on  him.  He 
pushes  down  on  the  earth,  which  reacts 
on  him,  pushing  up.  These  actions 
and  reactions  are  represented  in  Fig. 
12  by  the  arrows. 

We  may  consider  either  force  of  a  pair  as  the  action 
and  the  other  as  the  reaction.  Thus  in  the  present 
figure  we  consider  AI  to  be  the  action  of  the  man  on 
the  body  and  Ri  to  be  its  reaction  on  him.  We  are 
thus  viewing  the  action  from  the  standpoint  of  the  man. 
On  the  other  hand,  consider  the  crushing  effect  of  the 
weight  which  the  man  is  trying  to  support.  We  now 


150         THE   REALITIES   OF    MODERN   SCIENCE 

view  7£i  as  the  action  and  AI  as  the  man's  reaction. 
Which  one  of  a  pair  we  consider  the  action  is  merely 
a  matter  of  point  of  view. 

When  we  consider  the  forces  Ri  and  R2  of  the  figure 
we  notice  that  the  man  is  subjected  to  equal  and  op- 
posite forces.  This  pair  of  equal  and  opposite  forces 
also  constitute  a  stress.  Similarly  the  forces  AI  and 
A 2  constitute  a  stress.  The  man  is  exerting  a  pair  of 
equal  and  opposite  forces,  in  other~words,  a  stress. 
From  the  other  point  of  view  he  is  subjected  to  a  stress. 
In  all  cases  of  the  mutual  actions  of  bodies  stresses 
occur. 

For  bodies  at  rest  this  idea  is  very  evident  when  de- 
scribed in  terms  of  force  with  the  loose  meaning  which 
we  have  discussed  above.  Thus  a  body  at  rest  on  a 
table  acts  downward  on  the  table  and  is  supported 
by  the  upward  reaction.  When,  however,  a  body  in 
motion  is  considered,  the  idea  meets  an  instinctive 
objection  on  one's  part.  Thus  one  says,  "But  if  A 
pushes  B  away,  must  not  A  push  harder  than  B  ?  If 
the  reaction  is  just  equal  to  the  action,  why  doesn't 
it  just  balance  it  and  prevent  any  motion?" 

Consider  first  the  case  of  a  man  acting  on  a  bowling 
ball  instead  of  upon  another  man.  If  the  ball  is  of 
small  mass  it  does  not  require  as  great  an  action  upon 
the  man's  part  to  give  to  it  a  definite  acceleration.  In 
fact,  neglecting  air  friction,  the  force  which  he  is  con- 
scious of  exerting  depends  upon  the  mass  and  accelera- 
tion. These  determine  the  resistance  which  his  arm 
meets  in  swinging  the  ball.  If  it  was  a  gun  which  was 
shooting  the  ball,  the  reaction  of  the  ball  on  the  gun 
would  be  evidenced  by  the  recoil.  Why  is  there  no 


FORCE,  A  SPACE  RATE  OF  ENERGY  151 

recoil  to  the  man  as  the  ball  leaves  his  fingers?  Ob- 
viously because  he  serves  only  as  a  medium  which  trans- 
mits the  reaction  of  the  ball  to  the  earth. 

The  action  of  the  man  is  upon  the  ball  and  its  motion 
depends  only  upon  its  mass  and  the  force  which  he 
exerts.  Similarly  the  reaction  of  the  ball  on  the  man 
is  not  a  matter  which  has  a  further  concern  with  the 
motion  of  the  ball.  If  there  were  no  stress  between  the 
man  and  the  earth  his  momentum  would  be  equal 
and  opposite  to  that  of  the  ball.  Whether  he  takes 
up  the  momentum  himself  or  merely  transmits  it  to 
the  earth  does  not  alter  the  equality  of  the  action 
and  the  reaction. 

Now  consider  the  case  of  A  pushing  upon  B,  who  in 
turn  braces  against  the  ground.  As  long  as  B's  body 
will  stand  the  stress,  to  which  it  is  subjected  between  A 
and  the  earth,  there  is  no  motion.  When  it  will  no 
longer,  or  when  his  feet  slip,  then  he  moves.  So  far 
as  his  motion  is  concerned  he  is  to  all  intents  and  pur- 
poses a  bowling  ball.  His  reaction  on  A  makes  no 
difference  whatever  hi  his  own  motion,  although  it 
will  determine  the  motion  of  A  unless  the  latter  suc- 
ceeds in  imparting  it  to  the  earth.  That  this  is  so, 
is  readily  to  be  grasped  in  case  one  pictures  A  and  B  as 
engaged  in  their  struggle  on  a  polished  frictionless 
floor.  Then  if  A  grasps  B  he  may  thrust  the  latter 
away  from  himself.  As  he  does  so,  however,  there  is 
a  reaction  of  B  on  him.  The  result  is  that  both  A  and 
B  move  with  equal  momenta  but  in  opposite  direc- 
tions exactly  as  in  the  case  of  the  man  and  the  boat. 
If  the  floor  does  not  have  a  negligible  friction  the  heavier 
man,  of  course,  has  the  advantage,  for  he  will  transmit 


152         THE   REALITIES  OF    MODERN  SCIENCE 

to  the  earth  more  of  his  momentum  than  will  the  lighter 
man. 

Newton  demonstrated  the  equality  of  the  components 
of  a  stress  as  follows.  He  floated  two  bowls  in  a  large 
vessel  of  water.  In  one  was  a  magnet  and  in  the  other 
a  piece  of  iron.  The  magnet  attracts  the  iron.  The 
iron  reacts  on  the  magnet.  The  two  bowls,  therefore, 
float  together.  If,  however,  the  action  is  greater  than 
the  reaction  the  magnet  will  keep  pulling  the  iron  along 
and  the  combination  will  move  Conversely,  if  the 
reaction  is  greater  the  two  bowls  will  move  in  the  op- 
posite direction.  Since  neither  bowl  propels  the  other, 
action  and  reaction  must  be  equal  and  opposite. 

If  one  could  propel  the  other  there  would  be  a  viola- 
tion of  the  principle  of  the  conservation  of  energy. 
We  could  arrange  a  circular  water  path  and  allow  the 
magnet  to  propel  the  iron  around  this  path.  Connect- 
ing the  rim  of  a  wheel  to  the  magnet,  we  could  obtain 
useful  work.  The  result  would  be  a  "  perpetual  mo- 
tion" machine. 

Perpetual  motion  machines  are  those  which  will 
do  useful  work  without  requiring  that  at  least  an 
equal  amount  of  work  shall  be  done  upon  them,  that 
is,  machines  which  will  create  energy.  In  the  early 
days  of  experimental  science  there  were  many  seekers 
after  a  machine  of  this  type,  and  ingenious  were  the 
fallacies  by  which  they  persuaded  themselves  that  their 
designs  followed  known  laws  of  physical  science.  While 
the  search  was  fruitless,  the  experimentation  served 
some  purpose  in  advancing  science  just  as  did  the 
equally  fruitless  search  of  the  alchemists. 

To  that  master  philosopher,  Newton,  such  quests 


FORCE,  A  SPACE  RATE  OF  ENERGY  153 

seemed  futile,  for  he  stated  as  a  scholium  to  his  third 
law  what  we  know  to-day  as  the  "work  principle." 
What  he  said  may  be  expressed  in  our  modern  terms  as 
follows:  "In  any  machine  or  combination  of  machines 
the  work  of  the  acting  forces  is  equal  to  the  work  of  the 
resisting  forces."  If  we  understand  the  "work  of  re- 
sisting forces"  to  include  not  only  the  useful  work  but 
also  the  work  done  against  fractional  forces,  we  have  a 
statement  conforming  to  our  present  ideas  of  energy. 

To-day  the  goal  of  the  scientist  is  not  an  impossible 
perpetual  motion  machine  but  rather  the  efficient  utili- 
zation of  the  available  energy  of  the  universe  and 
that  increase  in  availability  which  may  follow  further 
knowledge  as  to  the  composition  of  matter.  Scientifi- 
cally the  aims  are  radically  different,  but  socially  they 
are  identical,  for  in  both  cases  the  ami  is  to  lift  man 
above  the  struggle  for  existence. 

To  understand  more  fully  the  present  limitations 
which  the  scientist  encounters  hi  making  available  for 
man  some  of  the  untouched  sources  of  energy  in  the 
universe,  we  need  to  consider  the  energy  of  molecules 
and  electrons.  In  so  far  as  the  energy  they  possess 
is  potential  we  are  in  ignorance  of  its  very  existence 
except  in  those  cases  where  we  have  already  learned 
to  release  it,  as  in  various  chemical  or  electrical  actions. 
In  so  far  as  it  is  kinetic  energy  we  may  measure  its 
amount  even  in  cases  where  we  do  not  as  yet  know  how 
to  utilize  it.  On  the  part  of  the  molecules  of  a  sub- 
stance such  a  measurement  is  possible  because  the 
energy  is  manifested  as  heat. 

The  kinetic  energy  of  translation  of  any  body  may 
be  expressed  in  terms  of  its  velocity  and  its  mass. 


154         THE  REALITIES  OF   MODERN  SCIENCE 

From  equation  (7)  we  see  that  the  energy  expended  in 
accelerating  a  body  of  mass  m  is  mas.  For  as  we  may, 
however,  substitute  v2/2  from  the  equation  v2  =  2as, 
which  was  derived  on  page  135.  Hence  the  energy  due 
to  translation  is  mv2/2,  where  v  is  the  velocity  at  the 
moment  of  our  consideration.  This  important  relation  is 
entirely  general  and  is  applicable  to  moving  molecules. 


CHAPTER  XIII 
MOLECULAR  MOTIONS  AND  TEMPERATURE 

IN  any  of  the  three  forms,  solid,  liquid,  or  aeriform, 
in  which  we  are  familiar  with  matter,  we  now  know 
that  the  molecules  are  in  rapid  motion.  In  solids 
under  normal  condition  the  molecules  do  not  move 
beyond  the  bounding  surfaces  of  the  body.  In  liquids 
moving  molecules  may  escape  from  the  surface,  as  in 
the  familiar  case  of  the  evaporation  of  water  from  an 
open  vessel.  The  molecules  which  have  escaped  move 
about  in  the  space  above  the  vessel,  unrestricted  except 
for  such  other  surfaces,  liquid  or  solid,  as  may  be  in 
the  neighborhood.  For  all  substances  which  are  in 
an  aeriform  state  such  unrestricted  motion  of  the 
molecules  is  characteristic. 

The  molecular  motions  of  the  latter  state  are  com- 
parable to  those  of  the  gnats  in  a  swarm,  such  as  one 
sometimes  meets  of  a  summer  evening  in  the  space 
below  a  street  lamp.  Their  motions  are  haphazard 
and  irregular.  Individual  gnats  approach  or  recede 
from  one  another,  flying  now  in  one  direction  and  now 
in  another.  They  picture  to  us  fairly  well  the  natural 
motions  of  individual  molecules.  The  entire  swarm 
may  be  given  a  motion  in  the  same  direction,  as  by 
waving  them  aside  with  a  paper.  Even  when  they 
have  such  a  motion  as  a  group,  they  still  have  their 

155 


156         THE   REALITIES  OF   MODERN  SCIENCE 

irregular  natural  motions.  In  the  case  of  wind  the 
motions  of  the  molecules  of  the  air  are  similar  in  that 
they  consist  of  these  natural  haphazard  motions  and 
of  a  forced  motion,  essentially  the  same  for  all  the 
molecules,  which  is  superimposed  upon  the  natural 
motion. 

If  one  thrusts  a  paper  into  a  swarm  of  gnats  he  will 
recognize  that,  at  whatever  angle  he  may  hold  it,  there 
will  be  about  as  many  gnats  striking  it  each  moment, 
for  at  any  instant  in  their  haphazard  motions  there 
are  about  as  many  flying  in  any  one  direction  as  in  any 
other.  Imagine  the  gnats  to  be  inclosed  in  a  box. 
There  should  be  about  as  many  striking  any  square 
centimeter  of  its  inner  wall  in  a  second  as  there  are 
striking  any  other  equal  area.  Their  impacts,  there- 
fore, should  exert  the  same  pressure  on  all  the  contain- 
ing walls.  That  a  large  number  of  such  small  blows 
produces  the  effect  of  a  steady  force  is  well  illustrated 
by  the  force  exerted  on  an  umbrella  by  the  raindrops 
of  a  heavy  summer  shower.  About  the  same  number 
strike  each  second,  and  the  umbrella  must  be  supported 
against  a  constant  force. 

Suppose  that  the  volume  of  the  box  is  halved  by 
pushing  the  cover  down  and  thus  crowding  the  gnats 
closer  together.  They  continue  to  fly  as  before,  but 
the  distance  between  top  and  bottom  is  only  half  as 
great,  and  therefore  they  require  but  half  the  time  to 
make  the  trip.  There  should  then  be  twice  as  many 
impacts  per  second  as  before,  and  the  pressure  on  the 
top  and  bottom  should  be  doubled. 

The  area  of  the  sides  of  the  box  has  been  reduced 
to  one  half  by  decreasing  the  height.  Since  for  each 


MOLECULAR   MOTIONS   AND   TEMPERATURE     157 

side  there  will  be  the  same  number  of  impacts  per 
second  as  before,  the  number  on  each  unit  of  area  and 
hence  the  pressure  on  the  sides,  also,  will  be. doubled. 
We  notice,  then,  that  as  the  volume  is  halved  the 
pressure  is  doubled;  in  other  words,  the  pressure 
exerted  by  the  moving  gnats  and  the  volume  of  their 
container  will  vary  inversely,  exactly  as  Boyle  found 
was  true  for  a  definite  volume  of  air. 

Suppose  that  we  keep  the  volume  constant  but 
introduce  an  additional  number  of  gnats.  If  there 
were  originally  N  gnats,  or  rather  let  us  say  molecules, 
and  this  number  is  increased  to  N'  then  the  number 
of  impacts  occurring  each  second  should  be  N'/N  times 
as  many  as  originally.  The  pressure  will  be  propor- 
tionately increased.  Thus,  if  it  was  initially  P  it  will 
be  P',  where  P'/P  =  N'/N.  Let  V  be  the  volume. 
It  has  not  changed  while  more  molecules  were  added. 
If  we  now  increase  the  volume  until  the  pressure  is 
restored  to  its  original  value  of  P  we  shall  find  the  new 
volume,  V,  to  be  greater  than  V  in  the  same  pro- 
portion as  P'  bore  to  P.  Hence  we  have  V'/V  =  N'/N. 
Therefore,  equal  volumes  at  equal  pressures  will  con- 
tain equal  numbers  of  molecules.  Or,  in  general,  if  the 
volumes  are  equal,  the  pressures  will  be  directly  as  the 
number  of  molecules.  Similarly,  if  the  pressures  are 
equal,  the  volumes  will  be  directly  as  the  number  of 
molecules. 

Consider  again  the  original  illustration  of  a  swarm  of 
gnats.  A  person  with  poor  eyesight  who  failed  to  see 
the  swarm  might,  nevertheless,  feel  the  pressure  of  their 
impacts.  Suppose  that  some  light  object,  larger  than 
a  gnat,  as  for  example  a  small  bit  of  very  thin  tissue 


158         THE   REALITIES  OF   MODERN  SCIENCE 

paper,  was  thrown  into  the  swarm.  It  wouldn't  fall 
very  quickly,  for  it  would  be  buffeted  first  one  way 
and  then  another,  now  up  and  now  down,  by  the 
impacts  of  the  various  gnats  which  collided  with  it. 
Such  a  piece  of  paper  might  be  made  so  small  that 
only  rarely  would  two  or  more  gnats  happen  to  collide 
with  it  simultaneously.  The  observer  while  unable 
to  see  the  gnats  would  then  see  this  paper  moving  in 
space,  irregularly  and  jerkily,  as  it  was  jostled  by 
individual  gnats. 

Now  this  is-  almost  what  actually  happens  in  the  case 
of  molecules.  The  human  eye  is  adapted  by  ages  of 
evolution  to  see  large  objects  which  are  of  interest  to 
human  beings.  Even  with  the  aid  of  the  best  possible 
microscope  it  can  never  see  directly  objects  smaller 
than  a  certain  definite  size  determined  by  the  color  of 
light  with  which  they  are  illuminated.  This  size  is, 
however,  much  larger  than  the  largest  known  mole- 
cule. In  1827  a  botanist,  Robert  Brown,  observed 
through  the  microscope  the  continual  movements 
of  some  minute  particles  which  were  suspended  in  the 
liquid  which  he  was  examining.  The  movements 
of  small  bodies  due  to  the  impacts  of  molecules  are, 
therefore,  known  to-day  as  "Brownian  Movements. " 
For  years  no  use  was  made  of  this  discovery,  although 
the  bacteriologists  learned  to  distinguish  between 
these  movements  of  small  suspended  particles  and 
those  of  the  bacteria  for  which  they  were  searching 
with  microscopes.  In  recent  years  many  very  im- 
portant ideas  as  to  molecular  movements  and  also  as 
to  electronic  movements  have  been  obtained  by  appli- 
cations of  this  phenomenon. 


MOLECULAR   MOTIONS  AND    TEMPERATURE     159 

Although  the  Brownian  movement  enables  scientists 
to  observe  by  their  effects  the  actions  of  individual 
molecules  and  has  thus  extended  our  knowledge  of 
molecular  physics,  we  shall  direct  our  present  attention 
to  the  average  behavior  of  a  large  number  of  mole- 
cules. In  other  words,  we  shall  treat  the  problem 
which  these  haphazard  motions  present  according  to 
what  is  called  the  "  statistical  method."  You  know, 
for  example,  that  if  you  flip  a  coin  it  is  just  as  likely 
to  be  " heads"  as  " tails."  There  are  only  two  pos- 
sible " events,"  as  they  are  called,  and  the  chance 
or  probability  of  one  is  the  same  as  that  of  the  other. 
In  the  same  way  the  chance  that  you  are  taller  than 
the  average  for  your  age  and  sex  is  equal  to  the  chance 
that  you  are  shorter  than  the  average.  If  one  assumes 
that  you  are  taller  his  probability  of  being  right  is  one 
chance  in  two,  or  one  half.  On  the  other  hand,  what 
is  the  probability  that  you  are  taller  than  your  two 
closest  friends?  There  are  three  possible  events, 
namely,  tallest,  taller  than  one,  or  shortest,  which  are 
all  equally  probable.  The  probability  of  any  one  of 
them  is  1/3. 

Consider  then  N  molecules  of  a  gas  in  a  rectangular 
box,  of  dimensions  a,  &,  and  c  centimeters.  (See  Fig. 
13.)  What  is  the  probability  that  any  particular 
molecule  is  moving  more  in  the  direction  of  the  a 
dimension  than  along  the  lines  of  b  or  c  ?  The  events 
are  all  equally  likely  and  the  probability  of  each  is  1/3. 
Of  the  N  molecules,  N/3  may  be  thought  of  as 
moving  in  the  direction  of  a. 

Not  all  the  molecules,  however,  are  moving  with 
the  same  velocity.  It  is  just  as  probable,  however, 


160        THE  REALITIES  OF   MODERN  SCIENCE 

that  any  individual  molecule  is  moving  with  a  higher 
velocity  as  that  it  is  moving  with  a  lower  velocity  than 
the  average.  We  may  therefore  deal  with  the  average 
velocity. 

What  is  the  effect  of  a  molecular  impact  against 
the  box?  If  the  molecules  were  inelastic  like  lumps 
of  putty  they  would  not  bound  back  from  the  walls 
of  the  box.  Their  kinetic  energy  would  be  imparted 
to  the  molecules  of  the  walls  and  they  would  cease  to 
move,  dropping  under  gravitation  to  the  bottom  of 
the  box.  The  pressure  they  exert  would  then  decrease 
to  zero.  If  they  are  elastic  they  bounce  back.  In 
order,  however,  that  the  pressure  shall  remain  constant 
as  time  goes  on  they  must  maintain  the  same  average 
velocity,  and  hence  on  the  average  they  must  bounce 
back  with  a  velocity  numerically  equal  to  that  of  their 
approach.  Otherwise  the  phenomenon  would  be  like 
that  taking  place  if  the  door  of  a  room  were  opened 
while  a  large  number  of  tennis  balls  were  thrown  in. 
The  door  might  then  be  closed  and  the  balls  left  to 
bounce  back  and  forth.  As  one  strikes  the  wall,  how- 
ever, it  does  not  bounce  with  its  original  velocity, 
because  some  of  its  kinetic  energy  is  converted  into 
molecular  motions  in  itself  and  in  the  wall.  The 
temperature  of  the  balls  and  the  walls  then  rises  as 
the  bulk  motion  of  the  balls  ceases.  That  the  kinetic 
energy  which  is  subtracted  from  a  moving  body  by  a 
collision  is  manifested  as  heat  is  well  illustrated  by  the 
rise  in  temperature  which  occurs  when  a  piece  of  metal, 
e.g.  a  nail,  is  pounded  with  a  hammer. 

If  the  average  velocity  of  the  rapidly  moving  mole- 
cules of  an  inclosed  gas  is  not  to  change,  two  conditions 


MOLECULAR   MOTIONS   AND    TEMPERATURE     161 

must  be  satisfied.  First,  there  must  be  no  subtraction 
from  their  energy  or  no  addition  to  it  by  the  walls  of 
the  box;  and  second,  there  must  be  no  similar  sub- 
traction or  addition  on  the  part  of  the  component  parts 
of  the  molecules. 

The  first  of  these  conditions  requires  that  on  the 
average  there  shall  be  no  interchange  of  molecular 
energy,  that  is,  heat,  between  the  walls  and  the  mole- 
cules. In  other  words,  they  must  be  at  the  same 
temperature.  This  does  not  mean  that  individual 
molecules  may  not  rebound  with  greater  or  less  velocity, 
but  means  that  on  the  average  the  value  of  the  kinetic 
energy  of  a  molecule,  represented  by  wv2/2,  must 
remain  constant.  Individual  molecules  may  lose  energy 
to  the  molecules  of  the  walls,  but  in  the  succeeding 
impacts  other  gas  molecules  must  receive  equivalent 
additions  of  energy  from  the  walls.  This  condition 
requires,  therefore,  that  on  the  average  the  kinetic 
energy  of  the  molecules  constituting  the  walls,  and 
hence  their  velocities  and  momenta,  must  be  such  that 
in  their  collisions  with  the  molecules  of  the  contained 
gas  there  shall  be  no  net  interchanges  of  energy. 

If,  however,  the  walls  are  not  at  the  same  temperature 
such  transfers  of  energy  will  occur  until  the  partition 
or  division  of  the  total  energy  possessed  by  the  two 
sets  of  molecules  satisfies  the  above  condition.  It  is 
by  such  exchanges  that  two  bodies  which  are  at  differ- 
ent temperatures  arrive  at  the  same  temperature, 
when  placed  in  contact,  one  cooling  and  the  other 
heating.  This  process,  incidentally,  is  usually  spoken 
of  as  a  transfer  of  heat  by  conduction. 

It  is  important  to  note  that  while  we  started  with 


162        THE   REALITIES   OF   MODERN   SCIENCE 

the  idea  of  average  velocity  what  we  really  are  con- 
cerned with  is  the  average  square  of  the  velocity,  since 
this  measures  the  average  kinetic  energy. 

Let  us  now  consider  the  second  condition  which  must 
be  satisfied  if  the  average  velocity  of  the  molecules 
before  and  after  mutual  collisions  shall  remain  constant. 
We  recognize  two  types  of  motion  which  the  atoms 
composing  a  molecule  may  have.  They  may  move 
with  reference  to  each  other,  just  as  two  persons  in 
walking  together  may  vary  somewhat  their  relative 
positions.  They  may,  however,  preserve  the  same 
relative  distances  apart  and  yet  rotate,  one  around 
the  other  or  both  about  some  common  point,  just  as 
two  dancers  in  waltzing.  They  may,  of  course,  have 
a  motion  which  is  a  sort  of  a  combination  of  these  two 
internal  motions.  The  pressure  on  the  walls  of  the 
container  is  due  to  the  motion  of  translation  which 
the  molecules  have  and  not  to  the  internal  motions 
of  their  component  atoms.  As  the  molecules  collide 
there  might  occur  transformations  of  energy  whereby 
these  internal  motions  gained  at  the  expense  of  the 
motion  of  the  molecule  as  a  whole.  We  all  have  seen 
illustrations  of  this  phenomenon,  as,  for  example,  two 
skaters,  colliding  at  high  speed,  may  be  set  spinning 
if  they  do  not  meet  head  on.  If  the  average  velocity 
of  the  molecules  is  not  to  be  altered  by  such  collisions 
then  there  must,  on  the  average,  be  some  definite 
partition  of  the  total  kinetic  energy  of  the  molecules 
between  translation  and  internal  motions. 

As  long  as  there  is  no  alteration  in  the  total  energy 
of  the  molecules,  such  as  would  occur  if  the  walls  of 
the  container  were  at  a  different  temperature,  the 


MOLECULAR    MOTIONS   AND    TEMPERATURE     163 

pressure  of  the  gas  remains  constant  unless  the  volume  is 
changed.  This  is  the  experimentally  observed  phenom- 
enon of  which  Boyle's  Law  describes  the  general  case. 
Hence  the  average  kinetic  energy  of  the  molecules 
must  remain  constant  and  also  the  average  internal 
energy  of  its  atoms.  The  molecules,  therefore,  must 
rebound,  either  from  mutual  collisions  or  from  im- 
pacts  against  the  walls,  with  the  same  average  (squared) 
velocity. 

We  are  now  ready  to  answer  the  question  of  page 
160  as  to  the  effect  of  an  impact 1  with  the  walls.  Dur- 
ing it  the  molecule  is  first  stopped  and  then  started  in 
the  opposite  direction  with  an  equal  velocity.  If  its 
mass  is  m  and  this  velocity  is  v  there  is  a  change  of 
momentum  of  mv  in  stopping  it  and  another  of  equal 
amount  in  imparting  to  it  the  opposite  velocity.  The 
total  change  hi  momentum 
accompanying  each  impact 
is  then  2mv. 

An  expression  may  now 
be  obtained  for  the  pressure 
which  these  molecular  im- 
pacts exert  on  the  end  be  of 
the  box  of  Fig.  13.  If  we 

multiply  the  change  in  momentum  per  impact  (i.e.  2mv) 
by  the  number  of  impacts  per  second  we  obtain  the  total 
change  of  momentum  per  second,  that  is,  the  force 
exerted.  Dividing  this  force  by  the  area  be  gives  force 
per  unit  area,  that  is,  the  pressure. 

1  Bernoulli!  in  1738  was  the  first  to  suggest  that  the  molecules 
of  a  gas  were  in  constant  motion  and  that  their  impacts  resulted  in 
a  pressure. 


164         THE   REALITIES  OF   MODERN   SCIENCE 

Let  v  represent  the  average  velocity  of  the  mole- 
cules in  the  direction  of  the  dimension  a  of  the  box. 
So  far  as  concerns  the  impacts  of  any  of  these  mole- 
cules on  the  end  be  it  is  evident  that  there  will  be  one 
impact  for  every  time  a  molecule,  moving  with  this 
velocity,  would  make  one  round  trip1  between  this 
end  and  the  opposite  one  That  is,  for  every  2a  cm. 
which  it  travels  in  the  a  direction  it  will  strike  the  end 
be.  Since  it  travels  v  cm.  per  second  it  will  make 
v/2a  impacts  per  second.  Since  there  are  N/3  mole- 
cules which  are  in  effect  traveling  in  this  direction 
there  will  be  a  total  of  Nv/Qa  impacts  per  second. 
The  force  exerted  on  be  is  therefore 


fe> 
_(2m,)or—  — 

Dividing  by  the  area  be  gives  the  pressure,  P,  as 

p_W_fmtf\ 
3abc\  2  ) 

in  which  the  product  abc  is  the  volume  V  of  the  con- 
tainer.    Hence 


We  have  thus  obtained  a  relation  which  indicates  that, 
provided  the  average  kinetic  energy  of  translation, 
7V(mt;2/2),  of  a  group  of  TV  molecules  is  not  altered,  the 
product  of  the  pressure  and  volume  is  constant.  In 

1  Actually,  of  course,  any  particular  molecule  may  collide  with 
another  and  bounce  back.  The  other  molecule  also  bounces  back. 
The  net  effect  so  far  as  concerns  impacts  on  the  walls  is  just  the 
same  as  if  the  molecules  had  passed  by  or  through  each  other. 


MOLECULAR   MOTIONS   AND    TEMPERATURE     165 

other  words,  we  have  obtained  an  expression  which 
states  the  facts  of  Boyle's  Law. 

The  molecules,  considering  their  motions  of  trans- 
lation only,  have  what  is  called  three  "  degrees  of 
freedom. "  A  railroad  train  has  only  one  degree  of 
freedom.  It  may  move  back  and  forth  along  only 
one  direction,  namely  that  of  its  track.  A  pedestrian 
has  two  degrees,  for  he  may  move  along  a  north  and 
south  direction  and  also  along  an  east  and  west  direc- 
tion. Having  these  two  degrees  of  freedom  he  may, 
of  course,  move  anywhere  in  the  plane  of  the  earth's 
surface  at  his  locality.  Thus,  if  he  moves  northeast, 
we  may  think  of  each  small  displacement  in  that 
direction  as  the  result  of  small  displacements  to  the 
north  and  to  the  east.  An  aviator,  on  the  other  hand, 
has  three  degrees  of  freedom.  He  is  not  restricted  to 
motion  hi  a  line  or  in  a  plane,  but  may  move  through 
space.  So  far  as  concerns  his  motion  to  any  point 
we  may  think  of  it  as  the  result  of  component  motions 
along  three  directions,  as  N-S,  E-W,  and  Up-Down. 

In  the  same  way  we  may  think  of  the  total  kinetic 
energy  of  the  molecules  of  a  gas  as  energy  of  motion 
in  each  of  three  rectangular  directions.  The  kinetic 
energy  due  to  motions  along  any  one  of  these  directions 
or  axes,  is  on  the  average  just  as  much  as  that  due  to 
motions  along  any  other.  In  other  words,  if  the  total 
kinetic  energy  of  translation  is  thought  of  in  terms  of 
the  degrees  of  freedom,  it  is  evident  that  it  should  be 
equally  divided  among  them.  That  is,  there  is  an 
equipartition  of  energy,  each  of  the  degrees  of  freedom 
having  one  third  of  the  energy. 

As  is  evident  from  the  equation  (1),  if  we  keep  the 


166         THE   REALITIES  OF   MODERN  SCIENCE 

volume  V  constant  and  vary  the  average  kinetic  energy 
of  the  molecules  the  pressure  varies  proportionately, 
that  is,  directly.  Similarly,  the  equation  shows  that, 
if  we  keep  the  pressure  constant,  the  volume  varies 
directly  as  the  average  kinetic  energy  of  the  mole- 
cules. Alterations  in  this  kinetic  energy  are  made  by 
adding  or  subtracting  energy,  that  is,  by  either  heat- 
ing or  cooling  the  body  under  consideration.  If  any 
body,  when  placed  in  contact  with  the  gas  (or  with 
the  thin  walls  of  its  container),  does  not  cause  a 
change  in  the  product  of  gaseous  pressure  and  volume 
there  is  no  net  change  in  the  average  kinetic  energy 
of  the  molecules  and  we  say  that  the  body  has  the  same 
temperature  as  the  gas. 

It  was  upon  exactly  this  principle  that  the  first 
thermometer  operated.  This  was  made  by  Galileo 
in  1597,  over  two  hundred  years  before  the 
kinetic  theory  of  gases  was  first  formulated.  The 
instrument  was  essentially  of  the  form  shown 
in  Fig.  14.  You  see  at  once  how  it  may  be 
used  by  the  substitution  method  to  indicate 
when  two  bodies  are  at  the  same  temperature. 
If  first  one  and  then  the  other  is  brought  into 
contact  with  the  bulb,  then,  the  fact  of  an  equal- 
ity of  temperature  will  be  indicated  by  the  same 
FIG.  14.  position l  in  both  cases  for  the  liquid  in  the  stem. 

A  possible  source  of  error  in  such  use  of  a  thermometer  may  be 
mentioned.  Suppose  the  two  bodies,  A  and  B,  are  very  different 
in  mass,  e.g.  that  A  comprises  a  smaller  number  of  molecules  than 
B.  It  requires  a  greater  addition  of  energy  to  B  than  to  A  to  pro- 
duce the  same  increase  in  the  average  kinetic  energy.  Body  B  has 
the  greater  "heat  capacity."  On  the  other  hand,  suppose  the 
atomic  structure  of  A  is  less  complex  than  that  of  B.  Because  of 


MOLECULAR   MOTIONS   AND    TEMPERATURE     167 

It  is  most  convenient  to  use  a  thermometer  as  a 
direct  reading  instrument  instead  of  by  the  sub- 
stitution method.  It  must  therefore  be  calibrated  by 
adding  energy,  just  as  the  spring  scale  described  on 
page  31  was  calibrated  by  added  weights.  But  what 
is  the  condition  which  we  are  to  call  zero?  Obviously 
the  condition  when  the  average  kinetic  energy  of  the 
molecules  of  its  gas  is  zero.  This  condition  means 
zero  average  velocity  of  the  molecules  and  from  equa- 
tion (1)  we  see  that  it  corresponds  to  the  condition 
when  the  pressure  exerted  is  zero. 

In  the  development  of  science  a  knowledge  of  ther- 
mometry  preceded  by  more  than  two  centuries  knowl- 
edge of  any  such  behavior  on  the  part  of  the  gas 
molecules,  as  that  which  we  have  been  describing.  Fur- 
thermore, even  to-day,  it  is  impossible  to  subtract 
all  the  energy  from  a  gas  and  thus  reduce  it  to  an 
absolute  zero  temperature.  In  the  earlier  years  there 
were  then  two  possibilities,  either  (1)  the  experimenter 
might  assume  that  the  lowest  temperature  he  was  able 

the  partition  of  the  energy  added  to  a  molecule  between  the  degrees 
of  freedom  of  translation  and  of  internal  motions  the  more  complex 
molecule  would  require  the  addition  of  a  greater  amount  of  energy 
to  produce  an  equal  increase  in  its  k.e.  of  translation.  In  other 
words,  the  heat  capacity  per  molecule  is  higher  for  B  than  for  A. 

Suppose  that  B  has  a  heat  capacity  very  much  greater  than  the 
thermometer.  If  it  is  at  a  higher  temperature,  it  will  need  to 
lose  but  little  of  its  energy  in  order  to  bring  the  average  k.e.  of  the 
molecules  of  the  thermometer  to  an  equilibrium  value.  On  the 
other  hand,  if  A  has  a  small  heat  capacity  the  amount  of  energy 
which  it  must  transfer,  to  raise  the  temperature  of  the  thermometer, 
may  well  result  in  a  decided  decrease  in  its  own  temperature.  When 
an  equilibrium  has  been  reached  between  A  and  the  thermometer 
the  indication  of  the  latter  represents  a  temperature  below  that 
which  A  originally  had. 


168         THE   REALITIES  OF   MODERN   SCIENCE 

to  obtain  by  the  best  means  at  his  disposal  was  the 
lowest  obtainable  by  any  means  and  thus  take  that 
temperature  as  the  zero  of  his  scale,  as  did  Fahrenheit, 
or  (2)  he  might  arbitrarily  assume  a  zero.  The  latter 
course  was  followed  by  Celsius,  who  in  1742  devised 
what  we  know  to-day  as  the  " Centigrade  scale.7' 

The  zero  chosen  by  Fahrenheit  corresponded  to  the 
temperature  reached  by  a  mixture  of  sal  ammoniac 
(ammonium  chloride)  and  melting  snow.  In  the 
Centigrade  scale  the  zero  is  the  temperature  of  melting 
ice  and  water.  As  a  matter  of  fact  it  was  Fahrenheit 
himself  who  had  made  this  selection  possible  by  show- 
ing, sometime  previous  to  Celsius'  choice,  that  the 
temperature  of  such  a  mixture  is  constant  as  long  as 
the  ice  is  not  entirely  melted.  The  difficulty  previous 
to  Fahrenheit's  demonstration  had  been  that  it  was 
known  that  water  could  be  cooled  below  this  temper- 
ature without  ice  forming.  This  is  true,  but  the  con- 
dition is  essentially  unstable,  for  if  a  small  bit  of  ice 
is  dropped  into  the  water  freezing  occurs  with  great 
rapidity. 

As  an  upper  point  on  his  scale  Fahrenheit  chose, 
unfortunately,  the  blood  temperature  of  the  human 
body.  He  then  divided  the  interval  into  96  equal 
divisions.  Celsius,  however,  selected  the  boiling  point 
of  water  at  atmospheric  pressure  and  divided  the 
temperature  interval  between  zero  and  this  upper  value 
into  100  equal  degrees. 

The  positions  of  the  liquid  in  the  stem  of  a  ther- 
mometer like  that  of  Galileo,  corresponding  to  the  two 
arbitrarily  assumed  but  easily  reproducible  tempera- 
tures of  the  Centigrade  scale,  may  be  marked  and  the 


MOLECULAR  MOTIONS  AND   TEMPERATURE     169 

intervening  portion  of  the  stem  divided  into  100  equal 
divisions.  Such  a  thermometer  is  not,  however,  as 
satisfactory  for  use  as  that  shown  in  Fig.  15.  In  this 
form 1  the  volume  of  the  contained  gas  is  kept  constant. 
As  the  temperature  changes  corresponding 
changes  occur  in  the  pressure  exerted  by  the 
gas  and  hence  in  the  pressure  required  to 
maintain  its  volume  constant.  The  volume 
is  controlled  by  the  mercury  in  the  flexible 
tube  and  the  pressure  exerted  on  the  gas  is 
dependent  upon  the  difference  hi  levels  of  the 
mercury  hi  the  glass  tubes  forming  the  exten- 
sions of  the  flexible  U  tube. 

To  calibrate  this  thermometer  the  bulb  is 
inserted  in  distilled  water  containing  chipped 
ice.  The  mercury  column  is  then  adjusted 
so  that  the  gas  has  a  definite  volume  when 
it  has  reached  a  temperature  equilibrium 
with  the  melting  ice.  Under  these  condi- 
tions let  the  pressure  be  denoted  by  P0.  The 
ice  bath  is  then  replaced  by  a  steam  bath 
and  the  pressure,  PIOO,  which  is  required  to 
reduce  the  volume  to  V,  is  noted.  The  change 
hi  pressure  which  has  been  observed  cor- 
responds to  100°  Centigrade. 

The  change  of  pressure  per  degree  Centigrade  is 
then  (Pioo— Po)/100.  The  fractional  increase  in  pres- 

1  Such  a  piece  of  apparatus  is  used  as  an  instrument  of  precision 
for  calibrating  the  more  convenient  mercury  and  alcohol  thermom- 
eters with  which  we  are  all  familiar.  The  latter  require  calibration 
because  equal  intervals  on  the  stem  do  not  correspond  exactly  to 
equal  increments  in  temperature. 


170         THE  REALITIES  OF   MODERN  SCIENCE 

sure  per  degree  represented  by  a  is  found  by  dividing 
by  PO  ,  thus 

_  PIOQ—  PO  so\ 

-  100  Po 

Careful  measurements  have  shown  the  numerical  value 
of  a,  the  "  pressure  coefficient  at  constant  volume/' 
to  depend  somewhat  upon  the  kind  of  gas.  Hydrogen 
has  therefore  been  adopted  as  the  standard  for  the  gas 
thermometer. 

Returning  to  equation  (2)  we  remember  that  the 


2    (  mv2  \ 
-      -- 


quantity  -N(  -^-  }  is  constant  if  the  temperature  is 
o     \   2i    J 

constant  and  that  we  decided  to  measure  temperature 
from  an  absolute  zero  corresponding  to  a  value  of 
Nmv2/2  equal  to  zero.  We  are  therefore  interested 
in  finding  where  our  arbitrary  reference  point  of  zero 
degrees  Centigrade  may  happen  to  be  located  in  this 
absolute  scale  of  temperatures. 

Let  us  represent  temperatures  measured  from  this 
absolute  zero  by  T.  Since  we  do  not  as  yet  know  the 
absolute  temperature  corresponding  to  0°  C.  let  us 
represent  it  by  T0.  In  terms  of  absolute  temperature 
a  temperature  of  t°  C.  is  expressed  as  t  =  T—T0,  since 
T=  T0+t.  The  value  of  the  pressure,  Pt,  corresponding 
to  t  is  expressed  in  terms  of  the  pressure  coefficient  and 
the  pressure  at  0°  C.,  namely  PQ,  as  follows: 

P,  =  P0(1+«0  (4) 

Substituting  for  t,  and  for  a  its  value  for  hydrogen  of 
0.00367  =  1/273  gives 


MOLECULAR   MOTIONS   AND    TEMPERATURE.     171 

When  the  absolute  temperature  is  zero  (i.e.  T  =  0)  the 
pressure  is  zero  (P,=0).  Substituting  these  values  in 
(5)  and  solving  for  T0  gives  TO  =  273.  The  zero  of  the 
absolute  scale  of  temperature  is  then  273°  below  the 
arbitrary  zero  of  the  Centigrade  scale.  In  other  words 
the  absolute  zero  is  -273°  C. 

This  absolute  zero  is  probably  never  to  be  reached 
experimentally,  although  in  recent  years  remarkably 
low  temperatures1  have  been  obtained.  As  the  tem- 
perature of  a  gas  is  decreased  the  molecules  may  be 
crowded  closer  and  closer  together,  until  they  are  so 
close  that  they  are  in  the  liquid  rather  than  in  the 
aeriform  state.  This  crowding  together  and  consequent 
liquefaction  may  not,  however,  be  brought  about 
merely  by  an  increase  of  pressure.  There  is  for  each 
gas  a  certain  definite  temperature  above  which  it  is 
impossible  to  produce  liquefaction  by  compression. 

The  liquid  gas  may  be  solidified  by  a  further  reduc- 
tion of  its  temperature.  The  temperature  at  which 
this  occurs  is  called  the  freezing  point.  We  see,  then, 
that  gases  like  hydrogen  or  oxygen  may  assume  any 
of  the  three  forms  solid,  liquid,  or  aeriform,  depend- 
ing upon  the  temperature  and  pressure  to  which  their 
molecules  are  subjected.  This  is  the  familiar  action 
of  water ;  but  in  the  case  of  water  the  freezing  point 
and  the  boiling  point  (that  is,  the  temperature  at  which 
steam  condenses  under  atmospheric  pressure)  are  at 
temperatures  within  the  range  which  we  meet  in  our 
daily  lives  instead  of  only  under  laboratory  conditions. 
In  fact,  from  our  ideas  of  the  molecular  construction 

1  The  gas  helium  was  liquefied  at  a  temperature  of  -271.°3  C., 
that  is,  within  two  degrees  of  the  absolute  zero. 


172         THE   REALITIES  OF   MODERN  SCIENCE 

of  matter,  we  see  that  what  we  have  been  accustomed 
to  consider  solid  substances  are  merely  substances  for 
which  the  melting  point  (i.e.  the  freezing  point)  is  well 
above  the  temperatures  which  we  meet  under  our  cli- 
matic conditions.  The  characteristic  of  being  solid 
is  not  one  of  the  substance  itself  but  is  dependent  upon 
the  physical  conditions  of  pressure  and  temperature 
under  which  it  is  at  the  time.  Similar  statements 
may  be  made  as  to  substances  which  we  are  accustomed 
to  consider  liquid.  Substances  which  we  know  as 
gases  are  of  course  those  for  which  the  boiling  point 
(i.e.  the  liquefaction  point)  is  well  below  ordinary 
temperatures. 


CHAPTER  XIV 

MOTIONS  OF  ELECTRONS 

THE  phenomenon  of  the  electrification  of  two  dis- 
similar substances  was  explained  in  Chapter  VIII  as 
due  to  a  redistribution  of  the  electrons.  The  electrical 
charges  thus  produced  might  therefore  be  expressed  as 
a  definite  number  of  electrons,  that  is,  measured  by 
counting  the  number  of  similar  particles  of  electricity 
which  were  added  or  subtracted.  Of  course,  the  actual 
operation  of  counting  would  be  impossible,  because  of 
the  size  and  the  large  number.1  During  the  120  years 
preceding  the  demonstration  of  the  existence  of  the 
electron,  knowledge  of  electrical  phenomena  and  of 
methods  for  their  measurement  had  developed  rapidly. 
To  make  such  measurements  units  were  adopted  which 
are  not  as  logical  as  would  be  the  electron  for  unit  quan- 
tity. When  the  electron  was  recognized  its  electricity 
was  measured  and  expressed  in  terms  of  a  previously 
chosen  unit.  The  units  adopted  are  simpler  than  the 
more  logical  unit  because  they  are  defined  in  terms  of 
magnitudes  which  are  directly  measurable  and  are  also 
of  convenient  amount. 

1  In  fact,  if  during  the  electrification  the  number  transferred  is 
such  that  as  we  separate  the  two  bodies  we  find  a  force  of  1  dyne 
when  they  are  1  cm.  apart,  we  know  to-day  that  about  2.1X109 
electrons  have  been  transferred. 

173 


174         THE   REALITIES   OF   MODERN  SCIENCE 

The  early  study  of  electricity  followed  two  lines  be- 
cause  the  investigators  wrongly  distinguished  between 
"  fractional"  electricity  and  "  galvanic"  electricity. 
Electrification  produced  by  friction  had,  of  course, 
long  been  recognized.  The  quantitative  law  for  the 
action  of  two  charges  produced  in  this  way  was  stated 
in  1785  by  Coulomb  in  a  form  similar  to  Newton's 
law  of  gravitation.  The  force  acting  between  two 
charged  bodies  is  proportional  to  their  charges  and 
inversely  as  the  square  of  the  distance  between  cen- 
ters. This  law  leads  to  a  unit  for  electricity  since  it 
may  be  written  as 

(1) 


where  qi  and  q2  are  the  two  quantities  of  electricity,  F 
is  the  force,  r  is  the  distance  (between  centers),  and  K 
is  a  factor  of  proportionality  depending  upon  the 
choice  of  units  and  upon  the  medium.  We  must  select 
some  medium,  as  for  example,  air,  and  let  K  be  unity 
for  this  condition.  Applying  to  equation  (1)  the 
method  of  Chapter  X,  it  appears  that  unit  electricity 
is  such  a  quantity  that  when  placed  in  air  at  a  distance 
of  1  cm.  from  an  equal  quantity  it  will  repel  it  with  a 
force  of  1  dyne.  This  is  now  known  as  the  electro- 
static unit  of  electricity. 

About  1780,  Galvani,  professor  of  anatomy  in 
Bologna,  observed  a  peculiar  phenomenon  in  connec- 
tion with  the  legs  of  some  newly  skinned  frogs  which 
were  awaiting  his  examination.  In  the  room  there 
was  a  machine  for  producing  electrification  continu- 
ously by  the  friction  of  two  dissimilar  substances. 
When  a  sufficient  charge  had  been  accumulated  a  spark 


MOTIONS  OF  ELECTRONS  175 

would  pass  through  the  air.  This  miniature  lightning, 
Galvani  noticed,  caused  twitchings  of  the  muscles  of 
the  frogs'  legs.  This  started  him  upon  a  series  of  ex- 
periments as  to  the  effect  of  electricity  upon  vital 
actions,  in  the  course  of  which  he  was  rewarded  by 
another  accidental  discovery.  Some  frogs7  legs,  hung  by 
copper  hooks  from  an  iron  railing,  convulsed  violently 
when  they  swung  into  contact  with  the  railing.  Guided 
by  this  he  studied  the  effect  further.  He  arrived,  how- 
ever, at  a  wrong  explanation  of  it,  assuming  that  at 
the  junction  of  the  nerve  and  the  muscle  there  was  a 
separation  of  electricities. 

It  remained  for  Volta,  a  professor  at  Pavia,  to  show 
about  1800  that  the  source  of  electrification  was  in  the 
dissimilar  metals  and  was  made  available  for  continuous 
effect  if  the  two  metals  were  separated  by  a  liquid  like 
salt  water.  From  this  accidental  start  there  was  de- 
veloped the  science  of  electricity  as  we  know  it  to-day. 
Men  have  forgotten  the  original  experiment,  which 
Galvani  had  in  mind  to  do,  but  the  by-product  has  had 
far  reaching  effect.  His  name  is  preserved  in  science 
in  the  word  " galvanometer,"  meaning  an  instrument 
for  metering  "galvanic"  or  " voltaic"  currents. 

Volta's  first  form  of  electric  battery  was  the  "pile," 
consisting  of  successive  layers  of  copper,  zinc,  and  wet 
cloth  in  the  order  named.  A  later  arrangement  was  a 
series  of  glass  vessels,  each  containing  salt  water  and 
plates  of  copper  and  zinc.  These  plates  did  not  make 
contact  inside  the  vessels  but  were  connected  outside 
them  from  copper  to  zinc  as  illustrated  schematically 
in  Fig.  16.  In  each  vessel  the  zinc  plate  was  found  to 
be  negatively  and  the  copper  plate  positively  electrified. 


176 


THE  REALITIES  OF   MODERN  SCIENCE 


The  series  connection  allowed  the  fact  to  be  more  easily 
observed  with  the  apparatus  then  at  the  disposal  of 


f 

r^  

f~                                              +1 

+   ^ 

16. 

^=--= 

m 

^^ 

^PJ 

=^£=z= 

r^§ 
I^Z 

z^r^ 
:^= 

-_=r^^^ 

Si 

EF= 

Cu     Zn 

Cu     Zn 
FIG. 

Cu      Zn 

scientists.  The  galvanometer,  of  course,  had  not  yet 
been  invented  and  the  forms  of  electroscopes  in  use 
were  not  particularly  sensitive. 

Volta's  contribution  to  science,  described  in  modern 
terms,  was  the  discovery  of  a  means  whereby  potential 
energy  of  chemical  separation  could  be  converted  into 
kinetic  energy  of  electrons.  A  definite  connection 
between  chemistry  and  physics  was  thus  indicated. 
Just  as  there  is  a  unity  to  physics,  which  was  formerly 
violated  by  the  attempt  to  classify  phenomena  under 
the  five  headings  mentioned  on  page  60,  so  there  is  a 
unity  to  chemistry  and  physics.  Words,  of  course, 
change  but  slowly  as  time  goes  on,  and  too  frequently 
establish  artificial  barriers  to  our  mental  development. 
In  the  formal  education  which  we  obtain  in  schools  and 
from  books  these  barriers  remain  even  longer  than  they 
do  in  the  education  of  our  other  daily  experiences.  In 
most  school  curricula  physics  and  chemistry  are  con- 
sidered so  separate  that  they  are  studied  in  different 
years  and  from  texts  which  carefully  refrain  from 
encroaching  on  each  others'  fields.  Nevertheless,  the 
subjects  have  an  essential  unity.  An  attempt  to  ap- 


MOTIONS   OF  ELECTRONS  177 

proach  one  at  a  time  instead  of  both  at  once  frequently 
leads  to  a  neglect  of  the  fundamental  realities  and  hence 
to  a  treatment  of  each  subject  as  a  group  of  apparently 
unrelated  phenomena. 

To  designate  the  portions  of  the  two  subjects  which 
are  of  importance  to  students  of  both  a  new  name  has 
come  into  use,  that  of  Physical  Chemistry.  This  treats 
of  the  portions  of  chemistry  where  energy  relations,  the 
behavior  of  molecules,  and  then-  electronic  composi- 
tion, are  most  evidently  involved.  As  yet,  however, 
this  name  and  the  unified  treatment  of  these  funda- 
mental relations  has  been  reserved  for  the  occasional 
advanced  student.  In  elementary  treatments  the  sub- 
jects are  still  separated. 

The  phenomena  of  which  Volta's  batteries  are  an 
illustration  classify  under  the  title  of  physical  chem- 
istry. From  the  standpoint,  however,  of  the  engineer, 
who  is  interested  in  effects  rather  than  causes,  Volta's 
contribution  to  science  may  be  described  as  the  dis- 
covery of  a  source  of  continuous  current  at  a  low  but 
fairly  constant  potential  difference. 

By  current  we  mean  the  time  rate  at  which  electricity 
is  transferred,  that  is,  the  number  of  electrons  per  sec- 
ond which  move  across  any  section  of  the  conducting 
path.  Thus  if  the  zinc  and  copper  plates  of  a  voltaic 
cell  are  connected  by  a  wire  a  stream  of  electrons  flows 
through  this  wire  from  the  negative  zinc  to  the  positive 
copper  plate.  Whatever  units  we  use  for  measuring 
the  quantity  of  electricity,  unit  current  will  be  flowing 
in  the  wire  when  electrons  are  transferred  at  the  rate 
of  one  unit  of  quantity  per  second.  For  example,  using 
unit  quantity  as  defined  by  reference  to  Coulomb's 


178         THE   REALITIES  OF   MODERN  SCIENCE 

Law,  we  obtain  unit  current  in  the  electrostatic  system 
as  1  e.s.  unit  of  quantity  per  second.     Symbolically 

i=Q/t.  (2) 

The  system  formed  by  the  plates  of  the  battery  has 
potential  energy  which  may  be  converted  into  kinetic 
energy  of  electrons.  A  difference  of  gravitational 
potential,  as  we  learned  on  page  108,  is  measured  in 
energy  per  unit  mass.  In  the  similar  case  of  the  cell 
it  would  be  logical  to  express  the  potential  difference 
of  the  plates  in  ergs  per  electron.  In  general,  a  dif- 
ference in  electrical  potential  is  expressed  as  so  many 
units  of  energy  per  unit  of  electricity.  Several  units 
are  therefore  possible,  depending  upon  the  choice  of 
units  for  energy  and  quantity.  In  the  case  of  the  unit 
called  the  volt  (after  Volta),  the  energy  unit  is  the 
joule  and  the  quantity  unit  is  the  coulomb.  The  lat- 
ter, named  after  the  scientist  mentioned  above,  is  the 
quantity  represented  by  three  thousand  millions  of  the 
electrostatic  units  which  were  defined  by  using  his  law. 
The  corresponding  current  unit,  the  coulomb  per  second, 
is  called  the  ampere. 

If  we  represent  potential  difference  by  E,  quantity 
by  Q,  and  energy  or  work  done  by  the  system  as  TF, 
we  have 

E  =  W/Q 

or        W=EQ  (3) 

as  the  defining  equation  for  E.     If  the  quantity  Q  is 
expressed  by  equation  (2)  as  it  we  may  write 

W=Eit.  (4) 

In  order  that  a  body  shall  fall  through  the  space 

intervening  between  the  points  of  higher  and  lower 


MOTIONS  OF  ELECTRONS  179 

gravitational  potential  it  is  only  necessary  that  it  be" 
released.  In  its  free  fall  it  acquires  kinetic  energy  which 
is  available  upon  impact  for  conversion  into  other 
forms  of  energy.  The  larger  portion  of  this  energy  is, 
of  course,  converted  into  heat,  that  is,  the  haphazard 
motion  of  the  molecules  of  the  partners  to  the  collision. 
Some  of  the  energy  is,  however,  transmitted  away 
through  space  in  the  form  of  a  sound  wave,  the 
mechanics  of  which  we  have  previously  considered. 

This  case  of  free  fall  is  one  which  is  possible  for 
electrons  only  under  rather  special  conditions.  We  ob- 
tain it  in  the  laboratory  by  using  an  evacuated  vessel 
containing  two  metal  plates  which  we  connect  to  the 
positive  and  negative  plates  of  the  battery. 

In  all  other  cases  with  which  we  have  to  do  the 
electrons  are  impeded  in  their  fall.  This  is  true  when- 
ever a  "  conductor "  connects  the  two  points  of  dif- 
ferent potential.  If  the  fall  is  impeded  the  energy  is 
subtracted  during  it  instead  of  entirely  at  its  end.  The 
difference  between  the  two  cases  is  much  like  that  of  a 
ball  which  is  either  dropped  from  the  roof  of  a  building 
or  allowed  to  roll  down  the  stairs.  In  the  latter  case 
its  descent  consists  of  a  large  number  of  free  falls 
through  short  distances.  It  never,  therefore,  acquires 
the  high  velocity  which  is  obtained  in  the  other  case. 
With  each  impact  the  kinetic  energy  acquired  since 
the  previous  impact  is  converted  into  heat.  This  is 
essentially  the  phenomenon  of  the  conduction  of  elec- 
tricity through  metallic  conductors. 

When  the  fall  of  electrons  is  not  free  but  takes  place 
through  a  gaseous  medium  an  impact  with  a  gaseous 
molecule  may  be  sufficient  to  shake  loose  or  otherwise 


180         THE   REALITIES  OF   MODERN  SCIENCE 

free  one  of  the  component  electrons  of  the  molecule. 
If  this  is  to  be  the  effect,  the  electron,  before  its  col- 
lision with  the  molecule,  must  have  fallen  through  a 
sufficient  difference  of  potential  to  acquire  the  amount 
of  kinetic  energy  necessary  for  disrupting  the  system 
of  nucleus  and  electrons  which  constitutes  the  molecule. 
When  an  electron  has  been  shaken  from  a  gas  molecule, 
so  that  it  is  free  to  pursue  an  independent  path  and  to 
have  an  individual  existence,  the  molecule  is  said  to  be 
ionized.  The  two  parts  which  are  thus  formed  are 
called  "ions,"  meaning  "goers,"  but  it  is  preferable  to 
call  the  portion  which  is  still  of  molecular  size  the  ion 
and  to  speak  of  the  other  moving  part  as  an  electron, 
since  that  is  what  it  really  is. 

As  we  shall  see,  the  electron  may  later  join  company 
with  a  neutral  molecule,  that  is,  one  which  has  not  been 
ionized.  In  this  case  also  we  would  speak  of  the  new 
combination  as  an  ion.  We  may  thus  have  ions,  pos- 
itive or  negative,  depending  upon  whether  they  are 
formed  from  a  neutral  molecule  by  knocking  off  an 
electron  or  by  combination  with  an  electron.  The 
electrons  and  the  ions  are  "goers"  whose  motion  is 
conditioned  by  the  systems  of  potential  energy  which 
they  form  with  the  positive  and  negative  plates. 

Now  in  air  at  ordinary  pressures  and  temperatures 
the  mean  free  path  1  is  comparatively  short.  A  moving 

1  The  distance  which  a  molecule  in  its  haphazard  motion  would 
travel  on  the  average  between  two  successive  impacts  with  its  fel- 
lows is  called  its  mean  free  path.  This  will  be  smaller  the  larger 
the  number  of  molecules  per  unit  volume,  that  is,  the  greater  the 
density  of  the  gas.  The  average  distance  through  which  an  electron 
may  move  between  successive  impacts  with  the  molecules  of  its  gase- 
ous path  will  depend  upon  the  mean  free  path  of  the  gas  molecules. 


MOTIONS  OF  ELECTRONS  181 

electron  may  therefore  not  acquire  sufficient  energy 
between  successive  impacts  to  admit  of  its  ionizing  the 
gas  through  which  it  bumps  its  way.  Its  ability  to 
ionize  depends  upon  its  acquiring  the  necessary  kinetic 
energy  during  a  motion  comparable  with  the  mean  free 
path  of  the  gas.  But  this  kinetic  energy  is  equal  to 
the  change  in  potential  corresponding  to  the  distance 
which  it  moves  along  the  path  between  the  two  plates. 
If  this  change  is  sufficient  then  ionization  will  occur. 

The  phenomenon  when  ionization  occurs  is  that  with 
which  we  are  familiar  on  a  large  scale  in  the  case  of 
lightning.  On  a  small  scale  it  is  illustrated  by  the 
spark  discharge  between  the  electrodes  of  a  machine 
such  as  Galvani  was  using,  or  in  the  discharge  between 
the  electrodes  of  a  so-called  induction  coil  such  as  is 
used  in  the  ignition  system  of  an  automobile.  In  all 
such  cases  the  discharge  starts  because  of  the  presence 
in  the  medium  of  a  few  electrons.  These,  by  their  col- 
lisions with  neutral  molecules,  produce  other  electrons 
and  also  positive  ions.  The  number  of  electrons  which 
are  traveling  toward  the  positive  electrode  therefore 
increases  rapidly.  The  positive  ions  which  are  formed 
by  the  collisions  naturally  move  toward  the  negative 
electrode,  where  they  combine  with  its  excess  electrons 
and  neutralize  not  only  their  own  deficiency  of  electrons 
but  also  the  excess  of  the  electrode. 

Such  recombination  may  also  take  place  en  route. 
In  fact,  during  a  continuous  transfer  of  electricity 
through  a  gas  it  is  possible  for  the  same  atomic  nucleus 
to  change  partners  several  times.  Thus,  after  a  re- 
combination with  an  electron  has  again  made  it  neutral 
and  while  pursuing  its  own  haphazard  motions,  it  may 


182         THE   REALITIES   OF    MODERN   SCIENCE 

be  struck  by  another  electron  traveling  at  such  a  speed 
as  to  combine  with  it  instead  of  again  disrupting  it. 
As  a  negative  ion  it  would  then  take  up  a  directed 
motion  toward  the  positive  electrode.  The  molecule 
has  now  in  its  group  an  extra  electron.  None  of  the 
electrons  are  therefore  as  firmly  held  as  if  the  normal 
number  had  not  been  exceeded.  An  impact  even  with 
a  neutral  molecule  may  sometimes  be  sufficient  to  jar 
off  an  electron.  Whether  the  electron  jarred  loose  in 
this  way  is  the  latest  addition  or  one  of  the  original  ones 
makes  no  difference  and  can  probably  never  be  de- 
termined, since  they  are  all  alike  anyway. 

Before  discussing  further  the  picture,  which  we  have 
just  obtained  as  to  the  mechanism  for  the  conduction 
of  electricity  through  gases,  it  is  well  to  note  that  it  is 
that  of  the  general  case.  If  all  the  molecules  of  the 
gas  are  removed,  leaving  a  vacuum,  the  current  can 
be  carried  only  by  electrons,  since  there  are  no  mole- 
cules to  form  ions,  and  hence  all  the  electrons  must  be 
supplied  and  released  at  the  negative  electrode.  If 
electrons  are  not  so  released  there  can  be  no  current. 
In  the  case  of  a  gas,  if  there  happen  originally  to  be 
no  free  electrons  between  the  electrodes,  there  can  be 
no  current,  unless  some  electrons  are  released  at  the 
negative  electrode.  If,  however,  the  potential  differ- 
ence is  sufficiently  high  the  presence  or  release  of  a  few 
electrons  will  result  in  ionization,  and  hence  in  the  self- 
perpetuation  of  the  supply  of  carriers. 

In  conduction  through  solids,  molecular  motion  is 
restricted  and  the  entire  transfer  of  electricity  is  due 
to  the  motion  of  the  electrons.  In  this  case  com- 
binations probably  occur  in  much  the  same  manner  as 


MOTIONS  OF  ELECTRONS  183 

in  a  gas,  but  the  combinations  are  prevented  by  the 
other  molecular  masses  from  moving  toward  the  elec- 
trodes. In  the  case  of  metals,  which  are  the  best  con- 
ductors, there  are  always  electrons  moving  about  through 
the  substance.  Although  the  number  per  cubic  cen- 
timeter is  very  large,  it  is  small  compared  to  the  total 
number  of  electrons  in  this  volume.  A  metal  then  con- 
sists of  a  large  number  of  neutral  molecules,  a  few 
positive  molecules  (they  can  hardly  be  called  "ions" 
since  they  cannot  "go"),  a  smaller  number,  perhaps, 
of  negative  molecules  and  a  number  of  free  electrons 
just  equal  at  any  instant  to  the  difference  between  the 
number  of  positive  and  negative  molecules. 

In  the  case  of  those  liquids  which  conduct  electricity 
(pure  water,  for  example,  does  not)  we  shall  see  later 
that  there  are  positive  and  negative  ions  but  no  free 
electrons.  The  ionization  of  liquids  is  not  the  result  of 
collisions  with  free  electrons,  since  there  are  none  pres- 
ent. It  is  in  the  nature  of  a  spontaneous  dissociation 
and  depends  only  upon  the  chemical  composition  of  the 
liquid.  Of  this  phenomenon  of  "electrolytic  dissocia- 
tion" we  shall  have  more  to  say  later. 

For  the  moment,  the  important  matter  is  to  obtain  a 
general  picture  of  the  conduction  of  electricity.  Con- 
duction occurs  as  the  result  of  the  motion  of  electrons. 
This  may  be  an  actual  motion  or  it  may  be  a  motion  of 
what  we  might  call  "certificates  of  electronic  indebted- 
ness." A  positive  ion  is  essentially  a  certificate  of 
electronic  indebtedness  which  may  be  transferred  at 
will  and  may  be  satisfied  at  any  point  in  the  universe 
where  there  is  an  excess  electron.  (Sometimes  the 
certificate  will  call  for  more  than  one  electron,  as  is  true, 


184         THE   REALITIES  OF   MODERN  SCIENCE 

of  the  ions  of  some  liquids.)  In  the  transfer  of  elec- 
tricity, therefore,  between  two  plates,  say  A  and  B,  of 
which  B  is  positive  and  A  negative,  it  makes  no  dif- 
ference whether  the  transfer  is  an  actual  one  of  electrons 
moving  from  A  to  B,  or  is  accomplished  by  the  motion 
of  certificates  of  electronic  indebtedness  from  B  to  A, 
or  in  part  by  each  method. 

In  the  case  of  conduction  through  a  vacuum,  or 
through  any  conductor  if  ionization  does  not  occur,  the 
transfer  is  entirely  the  result  of  motions  of  the  electrons. 
In  the  case  of  gases  the  transfer  is  partly  by  individual 
electrons,  partly  by  the  certificates  or  positive  ions, 
and  partly  by  electrons  which  are  combined  with  mo- 
lecular masses.  The  latter  are  truly  carriers  in  the 
sense  in  which  a  horse  is  a  carrier  of  his  rider,  and  they 
move  in  the  direction  in  which  the  added  electrons 
would  move  individually.  The  word  " carrier"  is, 
however,  generally  applied  to  both  positive  and  neg- 
ative ions.  In  the  case  of  liquids,  as  we  have  noted, 
the  transfer  of  electricity  is  brought  about  entirely  by 
the  motion  of  these  carriers. 

Of  course  the  carriers  may  not  share  the  burden  of 
transfer  equally,  one  kind  being  swifter  than  the  other. 
In  other  words,  the  positive  and  negative  carriers  of  an 
electrolyte  may  have  what  is  called  different  "  mobil- 
ities." The  same  is  true  of  the  carriers  of  a  gas.  A 
study  of  the  mobility  of  the  ions  formed  from  various 
gases  and  also  of  their  rates  of  diffusion  yielded  some 
of  the  earlier  determinations  of  the  value  of  the  charge 
carried  by  the  ions  and  hence  of  the  amount  of  elec- 
tricity corresponding  to  an  electron. 

So  far   we    have  considered   only   the   mechanism 


MOTIONS  OF   ELECTRONS  185 

whereby  charges  are  transferred  between  two  plates  or 
electrodes  which  are  maintained  at  different  potentials. 
We  have  not  as  yet  discussed  the  manner  in  which 
electrons  may  be  released  at  the  negative  electrode. 
We  have  seen  that  in  the  case  of  metals  the  electrons 
are  comparatively  loosely  held  in  the  atomic  or  molecu- 
lar structure  with  the  result  that  they  are  always  avail- 
able for  conduction.  Free  electrons,  which  may  migrate 
within  a  metal  body  from  one  point  to  another,  are  the 
cause  of  the  better  electrical  conductivity  of  metals 
and  also  of  their  better  heat  conductivity.  Those 
substances  which  are  good  conductors  of  electricity  are 
also  usually  efficient  in  transferring  molecular  energy 
from  a  point  of  high  temperature  to  one  of  low  tem- 
perature. 

Increased  temperature  of  a  metal  results  in  increased 
energy  on  the  part  of  these  free  electrons,  as  we  should 
expect  from  the  idea  of  the  equipartition  of  energy 
which  was  developed  in  the  preceding  chapter.  From 
the  standpoint  of  the  kinetic  theory  we  are  not,  there- 
fore, surprised  to  learn  that  there  comes  a  time,  as  the 
temperature  of  a  metal  is  increased,  when  some  of  the 
electrons  have  acquired  a  sufficient  kinetic  energy  to 
carry  them  beyond  the  influence  of  the  molecules  of 
the  metal J  itself.  In  other  words,  we  can  picture  to 
ourselves  a  phenomenon  of  the  boiling  of  the  electrons 
of  a  metal  quite  similar  to  the  boiling  of  a  liquid. 

Because  the  electrons  are  of  smaller  mass  than  the 
molecules  we  should  expect  a  very  pronounced  boiling 
of  electrons  long  before  the  temperature  of  the  metal 

1  The  phenomenon  is  essentially  similar  to  that  of  surface 
tension,  which  is  discussed  on  p.  221. 


186         THE  REALITIES  OF   MODERN  SCIENCE 

reaches  the  melting  point  of  the  substance  itself. 
This  phenomenon  is  made  of  practical  use  in  the  so- 
called  "  audions,  "  thermionic  devices  which  have  proved 
of  great  value  in  radio-communication. 

As  a  metal  is  heated  we  may  obtain  through  our 
physical  senses  an  indication  of  its  increased  electronic 
activity,  for  we  may  feel  the  heat  radiated  to  us  through 
space.  As  the  temperature  rises  still  further  we  ob- 
tain a  visual  indication  in  the  dull  red  color  of  the 
metal.  The  kinetic  energy  of  the  electrons  has  on  the 
average  been  increased  and  hence  their  velocity.  The 
frequency  with  which  an  electron,  although  bound 
within  an  atom,  may  move  back  and  forth  or  once 
round  its  restricted  path  is  thus  increased.  We  thus 
associate  the  radiation  of  heat  with  slower  vibrations 
than  correspond  to  the  radiation  of  light.  Not  all  the 
electrons  have  the  same  frequency,  since  not  all  would 
have  just  the  average  value  of  kinetic  energy.  The 
result  is  that  as  the  temperature  rises  some  of  the  elec- 
trons reach  the  frequency  at  which  they  radiate  red 
light  before  the  others.  As  the  temperature  is  still 
further  increased  the  average  frequency  rises  and  more 
electrons  emit  red  light.  Those  which  are  faster  than 
the  average  now  emit  a  yellow  light  while  the  slower 
ones  still  radiate  heat.  As  the  temperature  rises  the 
other  colors  of  the  spectrum  are  emitted  and  the  metal 
becomes  incandescent.1 

The  radiations  of  the  fastest  electrons  are  beyond  the 
range  of  visibility,  but  they  affect  photographic  plates 
and  may  otherwise  be  detected.  We  are  accustomed 
to  divide  up  this  range  of  frequencies  and  to  call  those 

1 A  more  exact  statement  is  beyond  the  scope  of  this  chapter. 


MOTIONS   OF  ELECTRONS  187 

which  are  too  slow  to  give  red  light  "infra-red"  and 
those  which  are  too  fast  to  give  the  violet  light  which 
lies  at  the  other  end  of  spectrum  "  ultra-violet  ' 
The  radiations  of  the  ultra-violet  range  have  only 
begun  to  be  investigated,  but  we  shall  find  several 
known  facts  of  considerable  interest. 

For  example,  the  ultra-violet  radiations  reaching  us 
from  the  sun  are  the  cause  of  those  chemical  transfor- 
mations whereby  the  leaves  of  plants  exposed  to  sun- 
light turn  green,  while  the  growth  beneath  the  ground 
remains  white.  Ultra-violet  radiations  if  not  suf- 
ficiently reduced  in  intensity  by  the  air  through  which 
they  travel  may  also  produce  severe  burns,  particu- 
larly in  the  inner  eye  of  a  human  being,  as  has  been 
recognized  by  workers  with  electric  arcs  or  oxyacety- 
lene  welding  outfits.  For  this  reason,  where  street 
railway  rails  are  being  welded  electrically,  goggles  are 
worn  by  the  workers  and  signs  are  displayed  advising 
the  passer-by  not  to  look  at  the  flame. 

For  our  immediate  purpose,  however,  the  importance 
of  ultra-violet  light  is  its  ionizing  effect.  Ions  are  pro- 
duced in  gases  exposed  to  light,  rich  in  ultra-violet 
radiations,  as  is  easily  verified  by  their  increased  ability 
to  conduct  electricity.  Thus  if  the  ah-  between  the 
charged  leaves  of  a  gold-leaf  electroscope  is  exposed  to 
ultra-violet  radiations  the  leaves  quickly  collapse. 
The  explanation  lies  in  the  formation  in  the  ah-  of 
positive  and  negative  ions.  Whether  the  leaves  were 
charged  positively  or  negatively,  there  are  thus  made 
available  carriers  of  the  opposite  kind,  which  may 
move  to  the  leaves  and  neutralize  their  charges,  allow- 
ing them  to  collapse. 


188         THE   REALITIES  OF   MODERN  SCIENCE 

These  ultra-violet  radiations  may  also  serve  to  shake 
electrons  loose  from  the  metals  on  which  they  fall.  In 
fact,  the  electrons  of  the  surface  of  the  metal  are  forced 
to  vibrate  with  the  same  high  frequency  as  do  the 
electrons  of  the  source  of  the  ultra-violet  light.  The 
phenomenon  of  this  radiation  of  energy  is  similar  to 
that  of  sound,  as  discussed  in  Chapter  VI,  in  the  one 
respect  that  energy  is  transferred  by  a  wave  motion 
from  a  vibrating  source  and  results  in  a  similar  vibra- 
tion of  a  distant  particle. 

It  is  immaterial,  so  far  as  concerns  merely  the  fact 
of  shaking  loose  an  electron,  whether  the  electron  ac- 
quires the  necessary  energy  indirectly  as  its  share  of 
the  total  increase  of  energy  received  by  the  entire  body 
in  being  heated,  or  directly  by  transmission  from  a 
distant  electron.  Of  course  the  electron  may  also 
receive  the  necessary  increase  of  energy  by  the  direct 
impact  of  some  molecular  mass,  as  in  the  case  of  the 
conduction  of  electricity  through  gases,  when  the  pos- 
itive ions  of  the  gas  collide  violently  with  the  negative 
electrode  toward  which  they  naturally  move. 

The  emission  of  light  which  accompanies  ionization 
is  well  illustrated  in  the  discharge  which  takes  place 


€ 


FIG.  17. 


in  a  partially  evacuated  tube  like  that  of  Fig.  17.  The 
appearance  of  the  discharge  depends  upon  the  mean 
free  path  of  the  gas  molecules  in  the  tube  and  hence 
upon  the  evacuation.  The  form  shown  in  the  figure 


MOTIONS  OF   ELECTRONS  189 

is  merely  typical  of  those  which  may  be  observed. 
Near  the  negative  electrode,  or  cathode,1  the  positive 
ions  are  ionizing  the  gas  and  combining  with  the  excess 
electrons  of  the  cathode,  and  a  violet  glow  results. 
In  the  region  of  the  so-called  " positive  column"  there 
are  striae,  indicating  the  successive  layers  of  the  gas 
where  ionization  and  recombination  occur.  When  the 
vacuum  of  the  tube  is  carried  further  a  condition  is 
reached  where,  because  of  the  small  number  of  mole- 
cules available,  but  little  ionization  occurs  and  the 
effects  are  due  largely  to  the  electrons.  These  fly 
away  from  the  cathode  in  radial  lines.  Their  existence 
was  first  noted  by  Crookes  in  1876,  who  spoke  of  them 
as  " radiant  matter."  A  brilliant  phosphorescence  in- 
dicates their  impacts  with  the  walls  of  the  glass  tube. 
The  fact  that  they  proceed 
radially  is  usually  illustrated 
by  interposing  a  piece  of 
mica,  as  the  Maltese  cross  of 
Fig.  18,  and  observing  that  it 
casts  a  well-defined  shadow 
within  which  there  is  no  phos- 
phorescence. Some  of  the  ex-  FlG-  18- 
periments  by  which  it  was  proved  that  the  phenom- 
enon was  one  of  corpuscles  rather  than  of  light  rays, 
that  is,  the  experiments  which  led  to  the  identification 
of  the  electron,  will  be  described  in  Chapter  XXII. 

1  Current  was  considered  to  flow  from  a  positive  to  a  negative 
electrode  before  the  electron  was  discovered.  It  was  therefore 
spoken  of  as  flowing  from  the  ''anode "  to  the  "cathode."  The  tube 
which  we  are  considering  has  had  associated  with  it  the  names 
of  various  scientists.  It  is  preferably  described  by  the  name  of 
either  Crookes  or  Geissler. 


190         THE  REALITIES  OF   MODERN  SCIENCE 

When  the  electrons  are  allowed  to  strike  a  piece  of 
platinum,  as  A  of  Fig.  19,  they  give  rise,  under  proper 
conditions  of  vacuum,  to  what  have  been  called  X-rays, 
since  their  discovery  by  Rontgen  in  1895.  Their  im- 
pacts with  the  platinum  "  anti-cathode"  result  in  very 

large  changes  in  their 
momenta,  since  they 
b_  are  moving  with  ve- 
locities nearly  that  of 
light,  for  which  the 
velocity  is  3  X1010  cm. 
per  sec.  Just  as  the 

periodic  vibrations  of  electrons  radiate  energy  so  the 
sudden  change  in  motion  sends  out  energy  in  radial 
lines.  Except  for  the  enormous  difference  in  the  velocity 
and  for  the  fact  that  the  medium  is  different,  the  phe- 
nomenon is  somewhat  similar  to  the  sharp  crack  which 
accompanies  the  blow  of  a  baseball  and  a  bat.  There 
travels  outward  through  the  ether  a  sudden  pulse.  When 
this  pulse  strikes  any  substance  it  is  so  violent  that  it 
affects  the  atoms  well  within  the  body  instead  of  merely 
those  at  the  surface  as  does  the  energy  of  light  and 
heat  waves.  In  other  words,  these  pulses  have  extreme 
penetration.  The  penetration  depends  upon  the  sub- 
stance, and  is  much  less  for  those  denser  substances 
like  the  metals  and  particularly  for  lead.  A  substance 
may  partially  absorb  the  pulses  and  hence  cast  a 
shadow  depending  upon  its  density  and  thickness. 
These  X-rays  therefore  admit  of  our  taking  shadow 
pictures.  The  shadows  may  be  observed  by  inter- 
posing the  object,  as  for  example  the  human  body, 
between  the  source  and  a  screen  of  zinc  sulphide,  or  a 


MOTIONS  OF  ELECTRONS  191 

photographic  plate  (since  the  pulses  will  affect  it  much 
the  same  way  as  does  light). 

The  mechanism  of  the  ether  whereby  energy  may 
thus  be  transmitted  through  space  from  one  electron  to 
another  is  not  as  yet  known,  although  we  know  many 
important  quantitative  laws  as  to  such  transmission. 
We  have  considered  the  unexplained  realities  which 
the  physicist  meets  to  be  matter  (i.e.,  electricity)  and 
energy.  The  ether  is  sometimes  taken  as  a  third  real- 
ity, but  its  explanation  will  probably  be  included  in 
that  of  the  other  two.  Thus,  when  we  know  why  an 
electron  and  an  atom  which  has  lost  an  electron  con- 
stitute a  system  the  potential  energy  of  which  tends  to 
decrease  as  the  parts  tractate,  we  shall  probably  know 
also  the  mechanism  of  the  transfer  of  energy  through 
space. 

The  fact  that  the  motion  of  an  electron  anywhere 
in  space  may  affect  the  motion  of  any  other  electron  hi 
space,  except  in  so  far  as  the  energy  which  the  first  is 
transmitting  is  absorbed  by  intervening  electrons,  is 
the  basis  of  all  so-called  "  electromagnetic  radiation " 
whether  manifested  to  us  as  heat,  light,  X-rays,  wire- 
less telegraph  radiations,  or  obscured  to  us  because  of 
our  insufficient  scientific  knowledge  and  hence  awaiting 
detection  by  future  scientists.  There  is  also  another 
whole  field  of  effects  which  moving  electrons  produce 
upon  other  electrons  in  their  immediate  neighborhood. 
These  are  the  basis  of  the  dynamos,  motors,  and  trans- 
formers with  which  the  electrical  engineer  deals.  They 
are,  however,  but  special  cases  of  the  more  general 
phenomenon  of  the  effect  of  an  electron  in  motion  upon 
other  electrons. 


192         THE   REALITIES  OF   MODERN  SCIENCE 

Before  considering  them  let  us  summarize  the  factors 
that  enter  into  the  conduction  of  electricity  between 
two  plates  or  electrodes  which  are  maintained  by  some 
means  or  other  at  a  constant  difference  of  potential. 
The  current  at  any  instant  depends  upon  the  number 
of  carriers  which  are  available  and  upon  the  velocity 
with  which  they  are  "falling."  The  velocity  will 
depend  upon  the  potential  difference  between  the  two 
electrodes,  but  the  number  of  available  carriers  will 
depend  upon  several  other  factors.  Thus  it  will  de- 
pend upon  whether  or  not  the  conducting  path  is 
ionized,  and  this  in  turn  depends  upon  what  the  medium 
is  and  upon  whether  or  not  the  potential  gradient  has 
been  sufficient.  If  the  path  is  ionized  the  number  of 
available  carriers  will  depend  upon  its  previous  history, 
for  it  was  the  carriers  which  were  present  in  the  pre- 
ceding instant  which  were  active  in  forming  those  now 
available.  The  number  of  carriers  will  also  depend 
upon  the  character  and  electronic  condition  of  the  neg- 
ative electrode  and  upon  the  energy  which  the  electrode 
or  the  conducting  path  may  be  receiving  in  the  form, 
for  example,  of  ultra-violet  radiations,  or  X-rays. 

In  the  case  of  metals,  however,  the  supply  of  carriers 
is  apparently  equal  to  any  demand  which  may  be  made 
and,  provided  the  temperature  of  a  metal  conductor, 
and  hence  the  mean  free  path,  is  maintained  constant, 
the  current  depends  only  upon  the  potential  difference. 
An  increase  in  the  current  is  accomplished  by  increas- 
ing the  average  velocity  with  which  the  electrons 
travel  toward  the  positive  plate  and  hence  the  num- 
ber per  second  which  crosses  any  area  of  the  conducting 
path.  Let  us  see  how  the  potential  difference  must 


MOTIONS  OF  ELECTRONS  193 

be  changed  to  produce  for  such  a  conductor  a  given 
change  in  current. 

If  the  average  velocity  is  doubled  the  same  number 
of  electrons  will  pass  a  cross  section  of  the  conductor 
in  half  the  time  previously  required ;  that  is,  the  cur- 
rent, or  rate  of  transfer  of  electricity,  is  doubled.  The 
kinetic  energy  of  the  electrons  is,  however,  quadrupled, 
since  it  varies  as  the  square  of  the  velocity.  This 
energy  is  dissipated  in  heat  in  the  circuit,  and  thus  we 
see  that  the  energy  required  to  force  a  current  through 
a  given  circuit  varies  as  the  square  of  the  current. 
This  is  known  as  Joule's  Law  and  is  usually  expressed  as 

W=Ri2  (5) 

where  R  is  a  factor  of  proportionality,  known  as  the 
resistance. 

Returning  to  the  numerical  problem  we  see  that  the 
energy  has  been  quadrupled  by  doubling  the  current, 
but  that  the  quantity  of  electricity  transferred  has 
only  been  doubled.  The  energy  per  unit  quantity, 
that  is,  the  potential  difference,  has  thus  been  doubled. 
The  current  and  potential  difference  are  therefore 
directly  proportional.1 

The  fact  that  the  current  is  directly  proportional  to 
the  difference  of  potential  is  usually  expressed  hi  sym- 
bols as 

E  =  Ri  (6) 

where  R  may  be  shown  to  be  the  same  factor  of  pro- 
portionality as  was  introduced  into  equation  (5)  above. 

1  This  is  also  evident  from  equation  (3),  namely,  W  =  EQ,  since 
if  W  is  made  four  times  as  large,  while  Q  is  made  twice  as  large, 
then  E  must  be  twice  as  large, 
o 


194         THE   REALITIES  OF   MODERN  SCIENCE 

This  relation,  which  we  have  reached  by  considering 
the  energy  of  the  falling  electrons,  is  known  as  Ohm's 
Law.  It  was  announced  in  1829,  years  before  the 
more  general  aspects  of  electrical  conduction  were 
recognized.  Covering  as  it  does  the  special  case  of 
conduction  through  metals  and  through  electrolytes, 
it  has  proved  of  great  value  to  physicists  and  engineers 
in  the  development  of  the  practical  utilization  of 
energy  in  the  form  of  electricity  in  motion.  To-day, 
however,  the  place  of  importance  which  it  holds  in 
most  elementary  presentations  exposes  students  to 
the  danger  of  considering  the  general  phenomenon  of 
conduction  as  an  exception,  instead  of  recognizing  that 
those  cases  for  which  the  law  holds  are  special  and 
simple  cases  of  the  general  phenomenon. 


CHAPTER  XV 

INTERACTIONS  OF  MOVING  ELECTRONS  IN  CON- 
DUCTING CIRCUITS 

WHEN  two  electrons  tractate,  the  space  rate  at  which 
their  potential  energy  varies  is  given  by  Coulomb's 
Law.  This  law,  however,  holds  only  for  electrons 
which  are  at  rest,  for  even  it  the  separation  is  constant 
the  potential  energy  of  the  system  will  be  constant 
only  if  its  parts  are  not  hi  motion.  This  phenomenon 
is  the  basis  of  those  so-called  electro-magnetic  methods 
for  converting  mechanical  energy  and  electrical  energy 
which  are  utilized  by  the  ''electrical  power"  industry. 

The  reasoning  which  we  shall  follow  hi  discussing 
this  behavior  of  electrons  may  be  illustrated  by  con- 
sidering a  somewhat  similar  problem  of  mechanics. 
A  body  resting  on  the  surface  of  the  earth  is  revolving 
about  the  terrestrial  axis  much  like  a  bit  of  mud  on  an 
automobile  tire.  The  tendency  of  the  mud  to  fly  off 
at  a  tangent  is  merely  another  instance  of  the  inertia 
which  Newton  recognized.  The  greater  the  kinetic 
energy  of  the  mud,  the  more  pronounced  is  this  tend- 
'  ency  and  the  greater  must  be  its  adhesion  to  the  tire, 
if  it  is  not  to  fly  off. 

The  body  has  potential  energy  with  the  earth  and 
also  its  own  kinetic  energy.  The  latter  is  divided  among 
two  degrees  of  freedom,  one  along  the  line  of  centers, 

195 


196         THE  REALITIES  OF   MODERN  SCIENCE 


i.e.  radial,  and  the  other  normal  to  this,  i.e.  tangen- 
tial. Suppose  the  body,  which  is  instantaneously  at  p 
of  Fig.  20,  follows  its  tangential  path  to  the  point  p'. 
In  this  position  its  motion  is  partly  away  from  the  earth, 
along  Op',  and  partly  at  right  angles  to  this  direction. 

There  is  now  kinetic  energy 
—  in  both  degrees  of  freedom, 
although  in  the  previous  posi- 
tion the  radial  k.e.  was  zero 
and  the  motion  was  entirely 
tangential.  In  departing  from 
a  circular  path  it  acquires 
radial  kinetic  energy,  and 
also  increases  the  potential 
energy  of  the  system.  (The 
tangential  k.e.  is  decreased.) 
FlG-  20-  The  change  in  the  p.e.  is  due 

to  the  increased  separation  from  the  earth,  but  that  in 
the  k.e.  is  due  to  the  new  angle  which  the  direction  of 
motion  makes  with  the  radial  line.  The  radial  k.e.  and 
the  p.e.  do  not  necessarily,  therefore,  increase  at  the 
same  rate  with  respect  to  the  space  over  which  the 
body  moves.  The  radial  energy  is  available  for  further 
outward  motion  and  if  it  increases  more  rapidly  than 
the  potential  energy,  the  body  will  continue  to  move 
away  from  the  earth.  Whether  or  not  it  does,  depends 
upon  the  total  kinetic  energy  and  hence  upon  how  fast 
the  surface  of  the  earth  is  rotating  about  its  axis. 

The  limiting  l  case  occurs  when  the  increase  in  radial 

1  Calculations  show  that  for  bodies  on  the  surface  of  our  earth 
this  limiting  value  would  be  reached  if  the  earth  should  revolve 
about  290  times  as  fast  as  at  present.  Bodies  on  the  surface  would 


INTERACTIONS  OF  MOVING  ELECTRONS       197 

k.e.,  in  moving  a  small  distance  like  pp',  just  equals 
the  increase  in  p.e.  Any  such  outward  motion  as 
that  represented  in  the  figure  would  then  be  impossible. 
On  the  other  hand,  even  if  not  supported,  it  could  not 
fall  toward  the  earth.  Under  this  condition  the  kinetic 
energy  of  the  rotating  body  would  have  made  unavailable 
the  potential  energy  which  it  has  by  virtue  of  its  position. 

Below  this  critical  speed,  that  is,  under  actual 
conditions,  some  of  the  gravitational  potential  energy 
will  be  unavailable.  Such  a  reduction  of  available 
potential  energy  should  be  most  pronounced  at  the 
equator,  where  the  surface  speed  is  greatest.  Hence 
"g, "  the  space  rate  of  change  of  the  available  poten- 
tial energy  per  gram,  should  be  less l  nearer  the 
equator,  as  experiment  shows  it  to  be. 

Suppose,  however,  that  we  had  been  born  on  an  earth 
which  was  not  revolving  and  that  we  had  calibrated  a 
number  of  spring  balances  after  the  manner  described 
in  Chapter  III.  Suppose  that  one  night  our  hypo- 
thetical earth  was  set  into  a  rotation  like  that  which  we 
now  experience.  All  of  our  spring  balances  would 
register  a  little  light.  If  then  somebody  told  us  that 
our  earth  was  revolving  we  might  say  that  such  a  rota- 
tion had  resulted  in  a  repulsion  being  exerted  between 
the  earth  and  all  bodies  on  its  surface. 

In  much  the  same  way  men  became  accustomed  to 
electricity  at  rest,  but  following  Volta's  work  they  awoke 
suddenly  to  a  world  hi  which  electricity  is  also  moving. 

then  not  show  the  phenomenon  of  weight  at  all.     If  it  should  re- 
volve faster  they  would  fly  off  into  space. 

1  This  change  in  "g"  is  in  addition  to  the  smaller  change  due  to 
the  increased  separation,  as  discussed  in  Chapter  XII. 


198         THE  REALITIES  OF   MODERN  SCIENCE 

They  had  learned  to  speak  in  terms  of  force  and  to 
recognize  that  like  charges  repel,  while  unlike  attract. 
A  body  which  did  not  test  as  being  either  positive  or 
negative,  that  is,  a  neutral  body,  contained  equal  and 
opposite  charges.  Such  a  body  might,  however,  serve 
as  a  conductor  of  electricity.  Two  parallel  conductors 
did  not  attract  each  other  electrically,  for  the  attrac- 
tion of  their  unlike  charges  was  balanced  by  the  repul- 
sion of  their  like  charges.  Nevertheless,  when  currents 
flowed  through  them  in  parallel  directions  they  trac- 
tated.  Therefore,  men  reasoned  that  electricities  in 
motion  attracted  if  going  in  parallel  directions.  When 
it  came  to  electricity  in  motion  they  forgot  Coulomb's 
law  and  treated  it  as  a  new  entity,  calling  it  "  voltaic 
electricity,"  building  up  the  subject  of  " electro- 
magnetism,  "  so  called  because  such  currents  also  affect 
magnets,  and  making  new  definitions  for  such  electrical 
magnitudes  as  current,  quantity,  and  potential  difference. 
Let  us,  however,  express  this  phenomenon  in  terms 
of  energy  and  electrons.  Consider  first  two  wires, 
say  A  and  B,  which  are  not  carrying  currents.  We 
analyze  this  system  into  four  component  systems, 
namely  those  comprised  by  (1)  the  free  electrons  of 
A  and  of  B,  (2)  the  positive  molecules,  (3)  the  elec- 
trons of  A  and  the  positives  of  B,  and  conversely, 
(4)  the  electrons  of  B  and  the  positives  of  A.  These 
potential  energies  exist  but  are  absolutely  not  avail- 
able, e.g.  systems  (1)  and  (3)  are  equal  in  their  space 
rates,  and  the  parts  of  (1)  pellate  while  those  of  (3) 
tractate.  If  we  could  isolate  one  system  we  should 
find  that  the  energy  which  we  made  available  would 
just  be  equal  to  the  work  which  we  had  done. 


INTERACTIONS  OF  MOVING  ELECTRONS        199 

If  then  the  two  wires  are  to  tractate,  energy  must  be 
given  to  the  system  from  some  outside  source.  If 
this  condition  is  brought  about  by  causing  parallel 
currents  we  may  justly  consider  it  to  be  due  to  a  de- 
crease in  the  availability  of  the  energy  of  system  (1), 
for  the  molecules  of  the  wires  do  not  move  more  dur- 
ing conduction  than  before  but  the  free  electrons  do. 
Individual  electrons,  of  course,  may  start  or  stop; 
some  may  move  with  velocities  higher  than  the  average 
and  some  with  less.  The  net  effect,  however,  is  just  the 
same  as  if  there  was  a  stream  of  electrons  formed  by 
a  definite  number  which  move  continuously  with  con- 
stant velocity.  The  available  potential  energy  of  a 
system  of  two  electrons,  moving  hi  parallel  paths,  is 
less  than  if  they  were  at  rest  by  an  amount  equal  to 
the  work  done  hi  setting  them  into  motion,  that  is, 
their  own  kinetic  energy.  Of  course,  because  of  the 
impacts  which  they  make  in  then-  travel,  energy  must 
constantly  be  supplied  equal  hi  amount  to  the  energy 
dissipated  in  heat  hi  the  conductors,  if  the  condition 
is  to  be  maintained. 

We  recognize,  then,  two  expenditures  of  energy  from 
the  batteries  which  cause  the  streams  of  electrons. 
The  first  of  these  is  equal  to  the  kinetic  energy  of  the 
electrons.  It  is  made  in  the  first  few  instants  after 
the  battery  is  connected.  The  second  is  the  constant 
energy  expenditure,  the  rate  of  which  is  expressed  by 
Joule's  Law.  The  current,  which  flows  in  the  first 
moment  or  so  before  a  steady  condition  is  established, 
depends  not  only  upon  conditions  hi  its  own  circuit 
but  also  upon  those  in  the  adjacent  circuit.  When 
the  battery  is  removed  from  the  circuit  the  moving 


200         THE   REALITIES  OF   MODERN   SCIENCE 

electrons  do  not  come  to  rest  immediately,  but  only 
after  they  have  expended  the  kinetic  energy  which  was 
originally  imparted  to  them. 

In  terms  of  energy  we  think  of  a  system  of  electrons 
as  having  potential  energy  which  is  reduced  if  they 
are  set  in  motion  along  parallel  lines.  The  reduction 
will  depend  upon  the  velocity  of  the  electrons.  When 
will  the  p.e.  be  zero,  and  can  two  electrons  ever  attract  ? 
The  answer  to  both  questions  comes  in  the  statement 
that  two  electrons  moving  with  the  velocity  of  light 
do  not  pellate.  The  velocity  of  light  is  that  with  which 
all  radiations  from  electrons  are  transmitted  through 
a  vacuum.  It  has  the  enormously  high  value  of  3  X 1010 
cm.  per  sec.,  and  appears  to  be  the  upper  limit  for 
velocity  in  our  universe.  Whatever  the  ether  may 
be  it  cannot  transmit  disturbances  faster  than  this.1 
Of  course,  where  electromagnetic  disturbances  are  im- 
peded in  their  passage  through  the  ether  by  the  pres- 
ence of  groups  of  electrons,  as  for  example  when  light 
passes  through  a  piece  of  glass,  the  velocity  may  be 
smaller  than  the  value  given  above.  The  velocity  of 
light  represents  tne  greatest  velocity  we  can  imagine 
in  a  universe  built  up,  as  is  ours,  of  electrons.  Two 
electrons,  then,  can  never  tractate  but  may  be  moving 
so  fast  that  they  do  not  pellate.  Thus  if  an  atom  could 
be  shot  off  through  space  with  the  velocity  of  light 
there  would  be  no  tractation  of  its  nucleus  and  elec- 
trons. There  would,  then,  be  nothing  to  prevent  the 

1  Media  which  transmit  energy  do  so  as  a  result  of  their  elasticity, 
the  velocity  of  propagation  in  each  being  determined  by  its  con- 
stants of  elasticity.  For  each  substance  there  is  therefore  a  definite 
velocity.  The  ether  behaves  like  an  elastic  medium  and  transmits 
all  disturbances  at  a  definite  rate. 


INTERACTIONS  OF  MOVING  ELECTRONS        201 

electrons  flying  apart,  but  they  would  have  no  tendency 
to  do  so. 

If  two  equal  circular  loops  are  placed  with  their 
planes  parallel  as  in  Fig.  21,  so  that  the  distance  between 
them  is  everywhere  the  same,  then  it  is  found  that  the 
force  of  attraction  is  proportional  to  the  two  currents 
and  inversely  as  the  distance  between 
them  and  also  directly  proportional 
to  the  length  of  the  current  path.  //  /  i  u 
If  one  of  the  loops  is  formed  of  U  I  /  // 
several  turns  of  wire,  since  the  same 
current  will  travel  through  each  turn, 
the  effect  will  be  increased  propor- 
tionately to  the  number  of  turns. 
If  the  current  hi  one  of  the  loops  is 
reversed,  then  there  is  a  repulsion 
equal  to  the  former  attraction.  For  convenience  hi 
determining  for  any  pair  of  such  circuits  whether  the 
action  is  one  of  repulsion  or  of  attraction  it  is  simplest 
to  view  both  circuits  from  some  point  which  is  not 
between  their  planes.  If  the  currents  are  in  the  same 
direction  there  is  attraction,  and  if  hi  opposite  direc- 
tions there  is  repulsion  Incidentally,  it  is  obvious 
that  in  a  coil  of  several  turns  the  various  turns  are 
mutually  attracted. 

Consider  the  case  of  a  fixed  coil,  ab,  and  a  movable 
coil,  cd,  shown  in  Fig.  22,  both  carrying  currents  in 
the  direction  of  the  arrows.  The  interactions  of  the 
electrons  hi  wires  a  and  c  cause  them  to  pellate  while 
wires  a  and  d  tractate.  The  result  is  that  the  coil  cd 
rotates  in  the  direction  of  the  heavy  arrow.  A  rule 
may  now  be  stated  for  the  rotation  of  a  movable  coil 


202         THE  REALITIES  OF   MODERN   SCIENCE 


with  reference  to  a  fixed  coil.  The  former  will  turn 
until  its  motion  has  reduced  the  potential  energy  of 
the  system  to  a  minimum,  in  which  condition  the  planes 
of  the  coils  and  the  currents  will  be  parallel.  In 
general,  coils  carrying  currents,  if 
free  to  move,  will  rotate  so  as  to 
make  their  currents  parallel  and 
will  also  tractate,  the  resulting 
motion  of  each  coil  being  a  com- 
bination of  the  two. 

If  the  coils  are  not  single  loops 
or  bunched  windings,  but  are  sole- 
noidal,  we  have  a  number  of  sys- 
tems of  single  loops.  Of  course, 
each  loop  of  a  coil  is  not  free  to 
turn  about  its  own  axis,  but  all 
must  turn  about  a  common  axis. 
The  motion  of  translation  is  the 
same  for  all  the  loops  of  the  same 
coil.  Each  of  these  systems  tends 
by  conversion  into  kinetic  energy 
to  reduce  its  potential  energy,  and 
the  space  rate  at  which  that  of 
the  entire  system  reduces  is  the  sum  of  the  rates  of 
the  component  systems ;  in  other  words,  the  force 
acting  is  the  sum  of  the  forces  due  to  the  several  loops' 
in  all  the  combinations  of  two  at  a  time  which  they  can 
form. 

In  the  solenoids  which  we  have  just  considered  the 
individual  loops  are  mechanically  connected  and  can- 
not revolve  about  axes  lying  in  their  own  planes. 
Imagine  one  solenoid  to  be  replaced  by  a  large  number 


FIG.  22. 


INTERACTIONS  OF  MOVING  ELECTRONS        203 

of  loops  which  are  free  to  rotate  about  individual 
axes  but  have  no  individual  freedom  of  translation. 
Suppose  also  that  each  loop  is  closed  on  itself  instead 
of  being  connected  in  series  with  the  others  and  that 
all  are  carrying  equal  currents.  What  would  be  the 
effect  if  such  a  group  of  loops  were  placed  along  the 
axis  of  a  solenoid  which  is  also  carrying  a  current  ? 
Each  loop  will  turn  so  as  to  reduce,  as  much  as  is 
possible,  the  potential  energy  of  those  similar  systems 
which  it  forms  with  the  loops  of  the  solenoid,  and  then 
a  further  reduction  will  be  occasioned  by  a  transla- 
tion of  the  group  as  a  whole  toward  the  solenoid. 

The  fact  that  a  current  is  flowing  in  each  loop  means 
merely  that  electrons  are  rotating  about  its  center. 
In  the  particular  case  under  discussion  such  a  rotation 
is  accomplished  by  the  motion,  in  the  circumferential 
path  formed  by  the  metallic  conductor,  of  its  free 
electrons.  The  molecules  of  the  metal  do  not  move, 
and  so  far  as  concerns  the  effect  in  which  we  are  in- 
terested the  only  function  the  metal  serves  is  to  keep 
the  electrons  in  the  path.  Suppose,  therefore,  that  a 
positive  nucleus  was  located  at  the  center  of  the  loop. 
It  would  serve  the  same  purpose.  If,  then,  there  are 
any  substances  the  atoms  of  which  are  formed  by 
electrons  rotating  in  a  plane  about  the  positive  nucleus, 
such  atoms  ought  to  act  just  like  the  single  loops 
which  we  have  been  discussing. 

What  would  be  the  test  ?  Obviously  if  we  bring  long 
cylindrical  pieces  of  different  substances  near  to  a 
solenoid  which  is  carrying  a  current  then  those  whose 
atoms  are  of  this  description  will  tractate  toward  it. 
In  the  case  of  iron  and  its  compounds  and  also  of 


204         THE   REALITIES  OF   MODERN  SCIENCE 

nickel  and  cobalt  this  effect  is  very  pronounced.  In 
most  other  substances  it  is  practically  negligible. 
The  substances  in  which  there  is  any  effect  whatever 
are  called  " paramagnetic/'  while  those  like  iron  are 
known  as  "  magnetic. " 

The  group  of  loops  which  we  have  described  is  a 
model  of  a  magnetic  body.  Since  the  loops  are  free 
to  turn,  thus  reducing  the  potential  energies  of  the 
systems  they  form  with  each  other,  and  since  the  greatest 
reduction  would  have  been  made  when  the  electrons 
were  moving  in  parallel  paths,  we  might  expect  that 
they  would  do  so  and  hence  that  in  a  magnetic  sub- 
stance the  molecules  would  naturally  have  assumed 
similar  orientations.  Such,  however,  is  not  the  case. 
The  molecules  have  haphazard  orientations,  as  may  be 
verified  by  placing  the  body  near  another  similar  body 
and  noting  that  there  is  no  attraction  or  repulsion,  as 
would  be  the  case  if  their  molecular  currents  were 
not  flowing  "  every  which  way."  The  explanation 
is  that  the  molecules  have  already  formed  themselves 
into  a  large  number  of  small  and  fairly  stable  groups. 

When  such  a  body  is  placed  near  a  loop  carrying  a 
current  some  of  these  groups  are  disrupted,  new  groups 
being  formed  with  more  molecules  oriented  in  the  direc- 
tion of  minimum  potential  energy  with  respect  to  the 
magnetizing  current.  Two  phenomena  are  now  possible 
when  the  magnetizing  coil  is  withdrawn  or  its  current  in- 
terrupted. Either  these  newly  formed  molecular  groups 
may  break  down,  due  to  their  interactions  with  adjacent 
groups,  or  many  of  them  may  be  so  stable  as  to  persist 
more  or  less  permanently.  In  the  latter  case  the  mag- 
netic body  is  said  to  be  permanently  magnetized  and  is 


INTERACTIONS  OF  MOVING  ELECTRONS        205 

called  a  magnet.  The  reason  why  two  magnets  attract 
or  repel  is  to  be  found  in  the  interactions  of  these 
oriented  loops. 

The  discovery  of  electricity  in  motion  by  Galvani 
and  that  of  a  source  by  Volta  was  followed  in  1820  by 
the  important  discovery  of  Oersted,  a  Danish  investi- 
gator. His  experiments  showed  that  a  magnet  was 
deflected  from  the  magnetic  meridian  by  a  conductor 
carrying  a  current  when  the  latter  was  placed  over 
the  needle  and  parallel  to  it.  The  direction  of  deflec- 
tion was  reversed  when  that  of  the  current  in  the  con- 
ductor was  reversed.  This  phenomenon  came  to  be 
explained  and  stated  in  terms  of  force  by  saying  that 
the  current  exerts  a  magnetic  force.  About  the  con- 
ductor there  is  said  to  be  a  magnetic  field,  that  is,  a 
region  in  which  a  magnetic  force  may  act. 

The  intensity  of  this  magnetic  field  at  any  point  was 
quantitatively  expressed  in  terms  of  the  force  which 
it  would  exert  upon  a  so-called  "unit  magnetic  pole" 
placed  at  the  point.  The  direction  of  the  field  was 
taken  as  that  in  which  a  "  north-seeking  "  pole  would 
move  under  the  action  of  this  force.  The  forces 
exerted  between  magnets  had  already  been  investi- 
gated by  Coulomb,  who  obtained  a  relation  identical 
in  form  with  that  for  electrical  charges,  namely, 

(1) 


where  m  is  the  pole  strength  and  v  is  a  constant  of 
proportionality.  Taking  /*  equal  to  unity  for  a  vacuum 
(or  air),  the  equation  defines  unit  pole  as  one  which 
will  exert  upon  a  like  pole  a  force  of  1  dyne  at  a  dis- 


206         THE  REALITIES  OF   MODERN  SCIENCE 

tance  of  1  centimeter.  The  defining  equation  for  field 
intensity,  H,  is  then 

#  =  -  (2) 

ra 

where  F  is  the  force  in  dynes  which  the  field  exerts 
on  a  pole  of  m  units. 

Experiments  during  the  years  immediately  follow- 
ing Oersted's  discovery  developed  the  fact  that  the 
magnetic  field  due  to  a  current  in  a  conductor  is  directly 
proportional  to  the  magnitude  of  the  current,  to  the 
length,  I,  of  conductor  if  it  is  everywhere  the  same 
distance,  r,  from  the  point  where  the  field  is  being 
measured,  and  inversely  as  r.  A  unit  for  current 
was  therefore  adopted  as  that  current  which  flowing 
through  1  cm.  of  the  arc  of  a  circle  1  cm.  in  radius 
would  establish  at  the  center  of  the  arc  a  field  of  unit 
intensity.  Such  a  current  would  then  exert  one  dyne 
of  force  on  a  unit  pole  at  the  center.  This  is  the 
electromagnetic  unit  of  current  in  the  C.G.S.  system. 

The  action  between  such  a  current  and  the  unit 
pole  is,  however,  in  the  nature  of  a  stress,  and  the  pole 
reacts  on  the  conductor  pushing  it  sidewise  with  an 
equal  and  opposite  force.  The  reaction  on  the  con- 
ductor is  therefore  1  dyne.  The  current  is  pushed 
sidewise  by  a  magnetic  field  of  unit  intensity,  for  the 
unit  pole  at  the  center  would  exert  1  dyne  of  force  on 
a  similar  pole  anywhere  on  an  arc  1  cm.  from  it.  The 
field  in  which  the  conductor  is  placed  is  unity  and  the 
length  of  the  conductor  1  cm.  If  field,  current,  or 
length  is  increased,  other  things  being  equal,  the  force 
exerted  on  the  conductor  is  proportionately  increased. 


INTERACTIONS  OF  MOVING  ELECTRONS       207 

That  is,  a  conductor  of  length  I,  placed  at  right  angles 
to  a  magnetic  field  H ,  and  carrying  a  current  i,  is  pushed 
sidewise  with  a  force  of  F.  Hence 

F  =  ilH  (3) 

is  the  most  convenient  form  for  the  electromagnetic 
relation  defining  current. 

While  this  equation  is  peculiarly  useful,  it  success- 
fully obscures  the  energy  relations.  All  ideas  of  the 
interactions  of  electrons  in  motion,  of  the  relative 
motions  of  the  two  bodies  containing  these  electron 
streams,  of  the  decreasing  potential  energy  which  is 
the  fundamental  cause  of  the  motion,  are  left  without 
either  expression  or  implication.  It  is  impraticable, 
however,  at  this  late  date  to  rewrite  this  defining 
equation  in  terms  of  energy  and  electrons. 

The  laws  for  the  motions  of  conductors  carrying 
currents  are  all  expressed  in  terms  of  the  positive 
carriers,  although  this  is  the  exact  reverse  of  the  actual 
mechanism  of  conduction  in  wires.  In  terms  of  elec- 
trons the  laws  might  be  expressed  as  follows:  The 
direction  of  a  magnetic  field  is  that  in  which  a  so-called 
north  pole  would  move.  In  any  coil  or  magnet  in 
which  the  direction  of  the  rotating  electrons  is  clock- 
wise the  field  is  directed  toward  the  observer.  Elec- 
tron streams  which  are  parallel,  tractate.  Streams 
which  are  in  opposite  sense  pellate.  The  motion  of 
a  conductor  at  right  angles  to  a  magnetic  field  is  at 
right  angles  to  both  the  direction  of  the  electrons  and 
that  of  the  field.  These  motions  are  related  as  the 
thumb,  center  finger,  and  forefinger  respectively  of 
the  right  hand. 


208         THE  REALITIES  OF   MODERN  SCIENCE 

In  1831  Michael  Faraday  performed  his  simple, 
illuminating  experiments  on  the  production  of  currents 
by  electromagnetic  induction.  These  phenomena  he 
pictured  in  terms  of  magnetic  lines  or  tubes  of  force 
which  extend  through  space  from  the  north  pole  of  a 
magnet  to  the  south  pole.  Such  lines,  also,  are  im- 
agined to  form  concentric  circles  in  planes  perpendicular 
to  a  conductor  carrying  a  current.  At  any  point  in 
space  their  number  is  considered  proportional  to  the 
strength  of  the  magnetic  field  and  their  direction  that 
of  the  field.  If  these  lines  are  assumed  to  have  two 
mechanical  properties  they  may  be  used  to  explain  all 
the  attractions  or  repulsions  of  magnets  and  coils. 
The  first  property  is  that  of  .contracting  indefinitely. 
The  second  is  that  of  exerting  a  pressure  on  each  other 
which  is  perpendicular  to  their  directions. 

Faraday's  discoveries  are  usually  expressed  by  saying 
that  a  relative  motion  of  a  conductor  and  a  magnetic 
line  induces  an  electromotive  force.1  An  e.m.f.  is  in- 
duced only  while  the  conductor  crosses  lines  of  force, 
and  its  amount  depends  upon  the  time  rate  at  which 
the  lines  are  cut.  If  the  conductor  forms  a  closed  cir- 
cuit, then  an  induced  current  flows  in  it  provided  that 
there  is  a  net  change  in  the  number  of  magnetic  lines 
threading  it.  It  makes  no  difference  whether  the 
cutting  is  occasioned  by  the  actual  motion  of  the  con- 
ductor relative  to  a  current-carrying  coil,  or  magnet, 
or  is  occasioned  by  changes  in  the  current  and  hence 

1  The  term  electromotive  force  originated  in  the  days  when  the 
causes  of  motions  were  sought  in  forces,  and  persists  to-day.  It 
is  essentially  synonymous  with  "  difference  of  potential  "  in  the  units 
of  which  it  is  measured. 


INTERACTIONS  OF  MOVING  ELECTRONS        209 

in  the  magnetic  field  of  a  coil  which  is  fixed  in  space 
with  reference  to  the  conductor.  In  every  case,  as 
was  first  stated  succinctly  by  Lenz,  the  direction  of 
the  induced  current  is  such  as  to  oppose  the  cause 
inducing  it.  This  is,  of  course,  an  electrical  statement 
of  the  principle  of  the  equality  and  opposition  of  ac- 
tion and  reaction  which  had  been  noted  by  Newton. 

If  in  one  of  two  parallel  loops  of  wire  a  stream  of 
electrons  is  started  in  a  clockwise  direction  there  is 
induced  in  the  other  loop  a  stream  in  a  counterclock- 
wise direction.  The  e.m.f.  in  the  second  circuit  exists 
only  while  the  current  in  the  first  coil  is  building  up, 
that  is,  only  while  there  are  more  electrons  starting  than 
stopping.  The  direction  of  the  current  in  the  second 
coil  is  obviously  such  that  the  coils  tend  to  move  apart. 
An  induced  current  of  the  same  direction  would  occur 
in  the  second  coil  it  it  were  moved  toward  the  first. 
Conversely  if  the  second  coil  were  moved  away  from 
the  first  the  induced  current  would  be  in  the  opposite 
direction,  that  is,  in  the  direction  to  oppose  the  motion 
inducing  it.  These  phenomena  are  usually  stated  in 
terms  of  magnetic  lines  of  force  and  the  motion  of 
positive  charges.  The  idea  of  energy  is  frequently 
obscured  thereby  and  the  motion  of  electrons  is  en- 
tirely neglected. 

In  discussing  energy  we  found  that  work  results  from 
a  relative  motion  of  the  parts  of  a  system  to  a  configu- 
ration of  smaller  potential  energy.  Two  parallel  con- 
ductors carrying  parallel  streams  of  electrons  tractate 
and  may  thus  do  work  in  the  same  manner  as  a  gravi- 
tational system.  We  therefore  say  that  the  conductors 
move  to  a  position  of  smaller  potential  energy,  The 


210          THE   REALITIES  OF    MODERN   SCIENCE 

potential  energy  of  the  system  is  not,  however,  as 
in  a  gravitational  system,  inherent  in  the  separa- 
tion of  the  two  bodies.  It  is  a  potential  energy  of 
ponderable  and  visible  masses  by  appearance  only. 
In  final  analysis,  it  is  a  kinetic  energy  of  the  electrons 
within  the  conductors,  for  unless  there  are  electron 
streams  there  is  no  potential  energy.  When  motion  is 
allowed,  external  work  is  done  at  the  expense  of  the  elec- 
trons. 

The  current  is  therefore  reduced.  This  is  not, 
however,  the  only  reduction  which  occurs.  Due  to 
the  relative  motion  of  the  two  conductors,  currents  are 
induced  in  them  in  such  directions  as  to  oppose  the 
currents  which  occasion  this  motion.  The  average 
velocity  of  the  free  electrons  which  are  taking  part  in 
the  conduction  is  therefore  reduced.  (An  induced 
current  occurs  whether  the  motion  inducing  it  is  caused 
by  an  expenditure  of  mechanical  energy  or  is  the  result 
of  the  mutual  attraction  of  the  two  parallel  currents.) 

If  the  electron  streams  are  not  allowed  to  decrease, 
but  are  maintained  at  the  expense  of  batteries  or  other 
sources  of  energy  connected  in  the  conducting  circuits, 
then  such  sources  must  supply  an  amount  of  energy 
equal  to  the  external  work  accompanying  the  motion, 
and  also  such  energy  as  is  required  to  restore  the  elec- 
trons to  their  former  velocities.  (We  are  neglecting 
any  stopping  of  the  electrons  due  to  collisions  in  the 
conductors  whereby  energy  is  dissipated  in  heat.) 

Let  us  state  this  phenomenon  in  a  slightly  different 
manner.  Two  conductors  carrying  parallel  currents 
possess  a  potential  energy  which  depends  upon  the 
currents  and  is  less  if  the  separation  between  the  con- 


INTERACTIONS  OF  MOVING  ELECTRONS        211 

ductors  is  less.  If  we  allow  them  to  tractate  we  find 
the  currents  reduced,  but  when  we  attempt  to  restore 
them  to  then-  former  value  we  have  to  supply  a  larger 
amount  of  energy  than  was  derived  in  external  work. 
In  restoring  the  currents  to  then*  former  value  we  have 
made  the  same  number  of  electrons  per  second  pass 
through  a  cross  section  of  the  conductor.  If  this 
represents  the  same  kinetic  energy  as  before  it  should 
have  been  necessary  to  supply  to  the  system  only  an 
amount  of  energy  equal  to  the  external  work.  We 
have  to  supply  as  much  again.1  It  therefore  appears 
that  the  same  number  of  electrons  moving  with  the 
same  velocity  now  possess  a  greater  kinetic  energy. 
This  conclusion  we  must  reach  if  we  accept  the  prin- 
ciple of  the  conservation  of  energy. 

We  are  accustomed  to  consider  the  kinetic  energy  as 
mv^/2.  Can  the  mass  of  the  electron  be  different  in 
the  two  cases?  This  brings  us  squarely  against  the 
question  as  to  what  we  mean  by  mass.  We  have  been 
using  mass  as  meaning  amount  of  matter,  but  we  have 
had  occasion  to  deal  either  with  electrically  neutral 
bodies  or  with  bodies,  like  the  pith  balls  mentioned  hi 
Chapter  VII,  which  are  composed  of  millions  and 
millions  of  neutral  molecules  and  comparatively  few 
excess  electrons.  We  have  measured  masses  in  terms 
of  the  standard  kilogram  by  a  comparison  of  the 
gravitational  potential  energies  of  the  unknown  and 
known  mass.  Or,  by  our  own  muscular  sense,  we  have 
compared  inertias  and  said  that  they  were  equal  when 
equal  forces  produced  equal  accelerations.  We  have  then 
said  that  the  bodies  of  equal  inertia  had  equal  masses. 
1  Cf.  footnote  of  p.  215. 


212         THE   REALITIES   OF   MODERN   SCIENCE 

The  only  conclusion  at  which  we  can  arrive  is  that 
part  or  all  of  the  mass  of  an  electron  is  what  we  might 
call  "  electromagnetic"  to  distinguish  it  from  the  sort 
of  masses  with  which  we  have  so  far  had  to  deal. 
If  we  say  that  all  of  the  mass  of  an  electron  is  of  this 
character,  that  is,  not  invariable  in  its  inertia,  then  we 
are  in  accord  with  the  modern  theory. 

Consider  now  the  conditions  of  the  conductors,  which 
we  were  discussing,  in  order  to  determine  what  factors 
influence  the  effective  inertia  of  an  electron.  We 
notice  that  the  electrons  are  in  motion.  We  also 
notice  that  the  kinetic  energy  of  two  parallel  electron 
streams  is  greater,  even  though  the  velocity  remains 
the  same,  when  the  streams  are  closer  together.  The 
inertia  or  mass,  as  we  prefer  to  say,  of  an  electron  de- 
pends, then,  upon  its  velocity  and  upon  the  parallel 
streams  of  electrons  in  its  neighborhood. 

It  has  been  found,  however,  that  for  velocities  which 
are  small  as  compared  to  that  of  light  the  mass  of  the 
electron  is  practically  constant.  If  the  mass  at  rest 
is  represented  by  m0,  while  that  for  motion  with  a 
velocity  of  v  cm.  /sec.  is  represented  by  m,  and  the 
velocity  of  light  by  c  cm.  /sec.  then 


—  v2/c2  " 

gives  values  for  m  which  have  been  checked  by  ex- 
periment for  speeds  from  about  0.3  to  0.8  that  of  light. 
For  experimental  purposes  electrons  moving  with  these 
high  velocities  are  obtained  from  X-ray  tubes  or  from 
radioactive  substances. 

For  velocities  below  about  0.1  c.  such  as  we  meet  in 


INTERACTIONS  OF  MOVING  ELECTRONS        213 

the  mechanical  problems  of  falling  bodies,  railroad 
trains,  aeroplanes,  the  transmission  of  sound,  or  even 
molecular  motions  in  gases  and  liquids,  we  may  con- 
sider mass  to  be  invariable.  For  such  velocities  as 
we  meet  hi  the  transmission  of  electricity  through 
metallic  or  electrolytic  conductors  the  mass  of  the 
electron  is  invariable  except  as  it  may  be  influenced  by 
the  presence  of  other  moving  electrons.  In  the  case 
of  electrons  emitted  by  radioactive  substances  we  can- 
not consider  the  mass  invariable,  for  it  will  depend 
upon  its  velocity. 

As  to  the  variation  hi  apparent  mass  and  hence  in 
kinetic  energy  due  to  the  motions  of  other  electrons, 
we  may  say  that  it  always  requires  more  energy  to 
impart  a  given  velocity  to  an  electron  if  it  is  set  into 
motion  parallel  with  that  of  another  electron,  but  less 
energy  if  its  direction  of  motion  is  parallel  and  opposite 
to  that  of  another  electron. 

In  establishing  a  current,  say  i,  in  a  conductor  of 
more  than  one  turn,  e.g.  in  a  solenoid,  the  mass  of  each 
electron  is  dependent  upon  the  existing  motion  of  all 
the  other  electrons  not  only  hi  its  own  loop  but  also 
in  all  the  other  loops.  The  calculation  of  the  increase 
in  mass  of  the  electron  under  consideration  is  therefore 
rendered  too  difficult  except  hi  the  case  of  a  few  cir- 
cuits of  simple  geometrical  form.  The  energy  re- 
quired to  establish  i  depends  upon  the  mass  and  this 
in  turn  upon  the  configuration  of  the  circuit  and  upon 
the  current.  For  each  electron  the  energy  will  be 
one  half  the  product  of  its  mass  and  the  square  of  its 
velocity.  In  a  circuit  like  that  of  a  metallic  conductor, 
where  the  number  of  free  electrons  available  for  transfer 


214         THE   REALITIES  OF    MODERN  SCIENCE 

is  constant,  the  current  will  depend  only  upon  the 
velocity  with  which  they  move.  In  other  words, 
velocity  and  current  are  proportional. 

The  total  kinetic  energy  represented  by  a  current, 
i,  is  therefore  proportional  to  i2.  Let  us  write  it  as 

K.E.=±Li*  (5) 

where  L  is  a  factor  of  proportionality  which  depends 
upon  the  mass  of  the  electrons  and  hence  upon  the 
geometrical  form  of  the  circuit.  The  factor,  L,  is 
usually  called  the  coefficient  of  self-induction  of  the 
circuit,  because  it  determines  the  currents  induced  in 
each  part  of  the  circuit  by  the  changing  electron  streams 
in  all  the  other  parts.  It  is  a  measure  of  the  retarda- 
tion or  negative  acceleration  of  the  electrons  occa- 
sioned by  their  interactions. 

The  self-inductance  takes  into  account  only  those 
interactions  occurring  between  the  electrons  of  its  own 
circuit.  If  there  is  in  the  neighborhood  of  a  circuit 
of  inductance  LI,  with  a  current  of  ii,  another  circuit 
which  is  carrying  a  current,  say  i%,  then  there  are 
possible  interactions  between  the  two  circuits  which 
may  either  increase  or  decrease  the  energy  required 
to  establish  the  current  ii  in  circuit  1.  Consider  first 
a  single  loop  and  let  its  inductance  be  L.  If  a  current 
of  /  is  flowing  in  it,  the  kinetic  energy  is  JL/2.  Now 
suppose  we  divide  this  circuit  lengthwise  of  the  con- 
ductor by  an  imaginary  plane  so  that  on  one  side  the 
current  is  ii  and  on  the  other  side  i%,  where  iz  =  I—  ii. 
The  energy  may  now  be  expressed  as 


K.E.  =  \LP  =  iL(ii+i,)»  =  JLti»+Ltit,+iLi;*     (6) 


INTERACTIONS  OF  MOVING  ELECTRONS        215 

Let  us  now  maintain  the  two  currents  by  two  separate 
batteries.  The  forms  of  the  two  circuits  obtained  by 
using  the  plane  are  identical  and  both  have  the  same 
inductance.  Let  us,  however,  represent  the  induc- 
tance of  each  circuit  by  its  proper  subscript.  The 
energy  represented  by  \Li?  above  then  becomes  \Lii-f  ; 
and  similarly  ^L2iz2.  These  are  evidently  the  energies 
which  the  circuits  would  possess  if  they  were  isolated 
in  space. 

But  what  about  the  energy  represented  by  Life  ?  Is 
it  intrinsic  to  circuit  1  or  to  circuit  2?  Obviously  it 
is  a  mutual  energy.  Let  us  therefore  write  it  as  Mi^ 
indicating  by  the  M  that  it  is  mutual.  The  total  energy 
of  the  system  may  now  be  represented  as 


K.E.  =  tLM+SLttf+Mite  (7) 

Which  circuit  has  contributed  this  mutual  energy? 
The  answer  is  :  both  circuits.  For  example,  if  we 
allow  the  current  to  become  established  in  circuit  1 
and  then  attempt  to  establish  the  current  ^  we  find 
that  the  acceleration  of  the  electrons  hi  circuit  2  has 
been  accompanied  by  a  retardation  of  those  of  circuit 
1.  Additional  energy,  therefore,  must  be  supplied  to 
circuit  1  to  maintain  the  velocity  of  its  electrons. 
The  value  of  M  may,  however,  be  negative,1  depend- 

1  If  motion  of  the  coils  occurs,  M  changes  in  value.  If  it  does, 
the  kinetic  energy  of  electrons  is  converted  into  kinetic  energy  of 
the  ponderable  coils  or  vice  versa.  In  the  first  case  (that  of  p.  211) 
it  may  be  shown  that  to  maintain  the  currents  the  battery  sources 
must  supply  an  increase  of  k.e.  of  electrons  equal  to  twice  the 
external  work  done  by  the  moving  coils.  In  the  second  case  the 
source  of  mechanical  energy  supplies  twice  the  decrease  in  k.e.  of 
the  electrons. 


216         THE   REALITIES   OF   MODERN   SCIENCE 

ing  upon  the  relative  directions  of  the  two  currents. 
If  it  is  negative  the  energy  expenditure,  required  to 
establish  in  the  circuits  the  corresponding  currents,  is 
less  than  would  have  been  required  if  they  had  been 
isolated. 

An  acceleration  of  the  electrons  in  one  circuit  pro- 
duces a  retardation  of  those  electrons  in  an  adjacent 
circuit  which  are  moving  in  a  parallel  direction.  The 
interactions  of  two  electrons  in  motion  therefore 
constitute  a  stress.  In  other  words,  Newton's  third 
law  applies  to  electrons  in  motion. 


CHAPTER  XVI 

THE  CONTINUITY  AND  CORRESPONDENCE  OF 
MOLECULAR  STATES 

THE  statement  that  matter  is  molecular  in  composi- 
tion implies  that  no  abrupt  change  takes  place  when 
a  body  of  matter  undergoes  a  change  from  one  to 
another  of  the  three  forms  —  solid,  liquid,  and  aeri- 
form. The  changes  in  state  are  successive  and  con- 
tinuous. Many  attempts  were  made  during  the  latter 
half  of  the  19th  century  to  formulate  a  relation  which 
would  represent  all  the  possible  states  not  only  of  a 
single  substance  but  of  all  substances  which  are  not 
mixtures  of  several  kinds  of  molecules. 

These  equations  were  usually  extensions  of  equation 
(2)  of  page  164  which  expresses  the  relation  between 
pressure,  volume,  and  kinetic  energy  of  translation, 
the  latter  being  a  measure  of  the  absolute  temperature. 
Let  w  represent  the  average  change  in  the  kinetic 
energy  of  translation  of  a  single  molecule,  when  the 
temperature  of  the  mass  is  changed  by  one  degree. 
Then  the  average  k.e.  of  translation  of  a  single  molecule 
in  a  mass,  which  is  at  an  absolute  temperature  of  T, 
will  be  wT.  For  N  molecules  the  energy  will  beNwT. 

Hence  PV  =  ~NwT  (1) 

o 

217 


218        THE  REALITIES  OF   MODERN  SCIENCE 

We  may  deal  always  with  the  same  number  of  mole- 
cules, whatever  the  substance  may  be,  by  taking  1 
mole  of  it  as  explained  on  page  85.  The  quantity 

Nw  is  then  independent  of  the  substance.     Hence  for 

2 
1  mole  let  us  write  5  Nw=R  where  R  is  constant.1 

o 

Hence  PV=RT  (2) 

This  equation  gives  the  relation  between  the  three 
variables  of  pressure,  volume,  and  temperature  which 
determine  the  " state"  of  a  gas.  For  example  suppose 

the  temperature  is  main- 
tained constant  at  a  value 
represented  by  T,  then  the 
points  corresponding  to  all 
the  possible  pairs  of  values 
may  be  represented  by  a 
hyperbolic  curve  like  that 


"  v  marked  PV=RT  in  Figure 
23.      If  the   temperature   is 

increased  to  TI  the  values  of  P  and  V  will  lie  on  a  new 
hyperbola  like  that  marked  PV=RTi.  These  curves 
are  called  isothermals. 

In  developing  equation  (2)  we  tacitly  assumed  that 
the  volume  actually  occupied  by  the  molecules  is 
small  as  compared  to  the  volume  V  in  which  they  are 

1  The  numerical  value  of  R  may  be  found  from  the  following 
data :  Under  standard  conditions  of  pressure  and  temperature, 
namely  76  cm.  of  mercury  and  0°  C.,  the  volume  of  2.016  gm.,  that 
is  1  mole,  of  hydrogen  is  22,410  c.  c.  Since  the  density  of  mercury  is 
13.6  and  g  is  980,  the  pressure  in  dynes  of  a  column  76  cm.  high  is 
13.6X76X980  or  1.013X106  dynes  per  sq.  cm.  The  temperature 
T  is  273.  Substituting  these  values  of  P,  7,  and  T  in  equation  (2) 
gives  R  as  83.2  X  106  ergs  per  degree. 


CORRESPONDENCE   OF   MOLECULAR   STATES      219 

contained.  This  assumption  is,  of  course,  not  valid 
at  high  pressures.  We  also  assumed  that  the  mole- 
cules of  a  gas  do  not  form  with  each  other  systems 
having  potential  energy,  and  hence  that  there  are  no 
attractions  or  repulsions  between  them. 

If  a  body  of  gas  for  which  there  are  no  molecular 
attractions  is  allowed  to  expand,  as  for  example  by 
pushing  a  piston,  it  will  do  work  at  the  expense  of  its 
molecular  k.e.  of  translation.  Its  temperature  will 
therefore  fall.  If  it  is  then  heated  until  its  tem- 
perature is  restored  it  will  require  from  the  source  of 
the  heat  an  amount  of  energy  just  equal  to  the  external 
work  done  in  the  expansion.  But  if  no  external  work 
is  done  during  the  expansion  then  no  energy  should  be 
required  to  maintain  the  temperature.  Thus  consider 
two  connecting  vessels,  A  and  B,  which  are  separated 
by  a  stopcock.  Let  A  be  filled  with  a  gas  at  a  high 
pressure  and  let  B  be  practically  a  vacuum.  Let  the 
system  be  immersed  in  a  vessel  of  water,  changes  in 
the  temperature  of  which  may  be  observed  by  a  sensitive 
thermometer.  After  the  temperature  has  become  that 
of  the  bath,  the  stopcock  is  opened.  The  gas  rushing 
from  A  into  B  does  no  external  work.  We  should 
therefore  expect  no  change  in  the  average  k.e.  and 
hence  no  change  in  the  temperature  as  indicated  by 
the  thermometer. 

On  the  other  hand,  if  the  molecules  form  with  each 
other  systems  the  potential  energy  of  which  increases 
with  their  separation  such  an  expansion  involves  an 
increase  in  their  p.e.  This  increase  can  be  obtained 
only  at  the  expense  of  the  kinetic  energy  of  the  mole- 
cules. The  temperature  of  the  gas  will  then  fall. 


220         THE  REALITIES  OF   MODERN  SCIENCE 

This  method  was  first  used  by  Joule,  who  found  no 
change  in  temperature.  The  conclusion  reached  above, 
as  to  the  equivalence  between  external  work  and 
energy  input  in  the  form  of  heat,  when  an  expanding 
gas  is  maintained  at  constant  temperature,  is  known 
as  Joule's  Law.  Later  and  more  precise  experiments 
showed  that  all  gases  cool  somewhat  upon  free  ex- 
pansion, except  that  for  hydrogen  there  is  an  anomalous 
warming  effect  at  ordinary  temperatures. 

If  Joule's  Law  does  not  hold  we  are  not  at  liberty  to 
use  the  equation  PV=RT  except  as  an  approximation. 
It  is  convenient,  nevertheless,  in  considering  gases  to 
discuss  the  " perfect  gases,"  that  is,  imaginary  gases, 
for  which  PV  =  RT  would  always  be  true.  Actual 
gases  depart  from  this  perfect  gas  relation  not  only 
because  the  molecules  occupy  some  space,  but  also 
because  the  compression  of  a  gas  is  not  due  solely  to 
the  externally  applied  pressure.  The  molecules  are 
brought  closer  together  by  the  tractation  which  ac- 
companies the  decrease  in  p.e.  of  the  systems  which 
they  form.  Except,  however,  for  temperatures  near 
that  at  which  the  gas  liquefies  the  perfect  gas  equation 
may  be  used  to  determine  the  condition  of  an  actual 
gas. 

In  liquids  the  molecules  are  very  close  together. 
A  molecule  in  the  body  of  a  liquid  moves  without  any 
effect  from  those  molecular  attractions  which  are  due 
to  its  potential  energy  with  the  other  molecules.  This 
is  because  the  other  molecules  are  disposed  about  it  in 
spherical  shells.  For  any  molecule  of  such  a  shell 
there  is  always  diametrically  opposite  another  mole- 
cule. With  these  two  the  particular  molecule  at  the 


CORRESPONDENCE  OF   MOLECULAR  STATES      221 

center  forms  two  balanced  systems  of  potential  energy. 
For  a  molecule  at  the  surface  as  in  Fig.  24  it  is  evident 
that  the  systems  are  unbalanced,  and  hence  it  would 
move  inward.  (The  systems  which  a  surface  mole- 
cule forms  with  other  molecules  on  the  surface  appear 
in  balanced  pairs,  and  there 
is  no  tendency  to  move 
along  the  surface.)  The 
p.e.  of  the  liquid  molecules 
is  therefore  decreased  by 

u  •  J  A!  FIG.  24. 

such    an    inward    motion 

until  the  surface  is  as  small  as  is  consistent  with  the 
volume  which  it  must  contain.  The  ratio  of  surface 
area  to  volume  thus  reduces  to  a  minimum,  and  hence 
liquid  drops  are  spherical  unless  affected  by  gravity. 
This  phenomenon  is  known  as  "surface  tension. " 

When  a  liquid  "wets"  a  solid  in  contact  with  it,  the 
molecules  of  the  liquid  and  those  of  the  solid  form 
systems  for  which  the  potential  energy  is 
greater,  other  things  being  equal,  than  it 
FIG  25  *s  f°r  ^e  systems  of  molecules  of  the  liquid 
alone.  The  liquid  molecules  tend  to 
spread  out  over  the  surface  of  the  solid,  reducing  as 
much  as  possible  their  separation  from  those  of  the 
solid.  Conversely  if  the  liquid  does  not  wet  the  solid 
it  is  unaffected  by  the  latter.  Thus  a  drop  of  mercury 
in  contact  with  clean  glass  assumes  the  oblate1  sphe- 
roidal form  shown  in  Fig.  25. 

At  the  critical  temperature  at  which  liquefaction  is 

1  The  drop  of  mercury  approaches  the  spherical  form  until  a  fur- 
ther increase  in  sphericity  would  cause  an  increase  in  gravitational 
p.e.  greater  than  the  decrease  in  molecular  p.e. 


222         THE   REALITIES  OF   MODERN  SCIENCE 

just  possible  the  phenomenon  of  surface  tension 
vanishes.  For  temperatures  below  this  value  the 
k.e.  of  the  molecules  is  insufficient  to  render  unavailable 
the  p.e.  and  tractation  occurs.  As  the  temperature 
is  decreased  from  the  critical  value  the  surface  tension 
increases,  although  at  first  very  slowly.  Now  water 
is  ordinarily  well  below  its  critical  temperature  of  360° 
C.,  and  in  it  therefore  we  find  a  very  noticeable  surface 
tension. 

Because  of  this  p.e.  between  molecules  and  also 
because  of  the  space  occupied  by  molecules,  the  perfect 
gas  equation  does  not  apply  to  an  actual  gas  except 
at  temperatures  well  above  the  critical  value.  The 
process  of  liquefaction  is  obviously  not  taken  into 
account  by  the  perfect  gas  equation,  for  the  molecules 
of  a  perfect  gas  could  be  compressed  and  cooled  in- 
definitely without  ever  acting  like  a  liquid,  since  they 
have  no  p.e.  It  would  be  convenient  if  we  had  an 
equation  of  state  which  was  applicable  not  only  to  actual 
gases  but  also  to  the  same  molecules  in  a  liquid  state. 
One  of  the  most  successful  attempts  to  develop  such 
an  equation  was  that  of  Van  der  Waals.  We  may 
consider  his  equation  to  be  formed  from  the  equation 
PV=RT,  by  the  introduction  of  two  correction  factors. 

The  first  of  these  is  introduced  to  take  into  account 
the  fact  that  the  volume  V  in  the  above  equation  is 
too  large  by  an  amount  dependent  upon  the  actual 
space  occupied  by  the  molecules.  Let  us  represent 
this  amount  by  b  and  hence  substitute  V— b  for  V.  The 
second  correction  factor  takes  into  account  the  p.e. 
of  the  molecules.  Because  this  tends  to  decrease  the 
molecules  tractate  and  the  effect  is  the  same  as  if  the 


CORRESPONDENCE  OF   MOLECULAR  STATES      223 

pressure  was  greater.  The  pressure  P  in  the  gas 
equation  is  therefore  too  small  by  an  amount  which 
depends  upon  the  separations  of  the  molecules,  being 
inversely  as  the  square  of  the  volume.  For  P  we 
write  the  expression  (F+a/F2).  Making  these  sub- 
stitutions, we  have  the  equation  of  Van  der  Waals 

(P+a/V*)(V-b)=RT  (3) 

or  multiplying  out 


The  values  of  a  and  6  may  be  determined  by  sub- 
stituting in  this  equation  the  known  value  of  R  and 
experimentally  determined  values  of  P,  V,  and  T. 

Equation  (4)  evidently  states  the  physical  relation 
by  which  we  should  expect  to  calculate  the  volume  V 
of  a  molecular  mass  upon  which  the  pressure  is  P  and 
of  which  the  absolute  temperature  is  T.  It  is  a  gen- 
eral theorem  of  mathematics  that  there  are  three  roots 
for  an  equation  in  which  the  highest  power  of  the 
unknown  magnitude  is  a  cube.  Since  V  enters  as  a 
cube  we  expect  to  obtain  three  numerical  values  for 
it.  But  can  a  molecular  mass  under  these  conditions 
fill  three  different  volumes?  If  we  consider  V  in  the 
perfect  gas  equation  to  be  the  unknown  there  is  only 
one  possible  value  which  it  may  have  for  any  assigned 
values  of  P  and  T.  According  to  equation  (4),  under 
the  same  conditions  of  pressure  and  temperature  there 
appear  to  be  three  different  volumes,  any  one  of  which 
the  molecules  might  occupy  equally  well,  so  far  as 
concerns  the  physical  conditions  represented  by  the 
equation. 


224         THE  REALITIES  OF   MODERN  SCIENCE 

We  can  conceive  of  three  such  volumes  if  we  take 
into  account  that  the  molecular  mass  may  be  either 
a  gas,  a  liquid,  or  part  liquid  and  part  gaseous.  When 
a  gas  changes  into  a  liquid  we  say  it  condenses,  and 
conversely  when  a  liquid  changes  into  a  gas  or  vapor 
we  say  it  boils.  This  reversible  phenomenon  we  shall 
need  to  discuss  before  considering  further  the  roots  of 
Van  der  Waals's  equation. 

The  temperature  at  which  this  phenomenon  occurs 
is  the  boiling  point.  The  boiling  point  is  raised  if  the 
liquid  is  subjected  to  an  increased  pressure.  Water 
boils  at  100°  C.  under  a  pressure  of  76  cm.  of  mercury. 
In  mountain  districts,  the  pressure  being  less,  boiling 
occurs  at  correspondingly  lower  temperatures.  In  a 
boiler  of  a  steam-engine  plant  the  boiling  point  of 
water  may  be  very  much  higher  than  100°  C.  At  the 
boiling  point,  however,  the  pressure  exerted  by  the 
water  vapor  molecules  is  always  just  equal  to  that  of 
the  air  or  vapor  molecules  above  them.  If  the  mole- 
cules are  not  allowed  to  escape  from  the  boiler,  as  is 
the  case  in  " getting  up  steam,"  the  increase  of  mole- 
cules in  the  space  above  the  liquid  results  in  an  increase 
in  the  pressure  against  which  those  still  in  the  liquid 
must  escape.  They  must,  therefore,  have  a  higher 
kinetic  energy,  and  when  they  do  escape  they  increase 
still  further  the  pressure  exerted  upon  the  liquid  by 
its  own  vapor.  The  pressure  of  the  steam  (and  water) 
thus  rises.  When  it  reaches  the  desired  value  steam 
may  be  withdrawn  to  run  the  engine,  but  the  molecules 
thus  withdrawn  must  be  continuously  replaced  by  others 
with  the  same  average  k.e.  or  the  pressure  will  fall. 

If  the  pressure  exerted  on  a  body  of  vapor  in  contact 


CORRESPONDENCE   OF   MOLECULAR   STATES      225 

with  its  liquid  is  increased  there  will  be  a  condensation 
unless  the  molecular  k.e.  is  correspondingly  increased, 
that  is,  unless  the  temperature  of  the  liquid  (and  vapor) 
is  raised  to  the  new  boiling  point.  Now  we  have  seen 
that  when  the  molecules  of  a  substance  are  in  the  liquid 
state  the  potential  energy  of  the  systems  they  form 
with  each  other  has  been  reduced.  Obviously  then, 
if  an  attempt  is  made  to  increase  the  pressure  exerted 
upon  a  liquid  and  its  vapor,  the  condensation  which 
occurs  will  cause  a  reduction  of  this  molecular  potential 
energy.  What  becomes  of  the  energy  thus  released? 
It  is  available  for  increasing  the  molecular  kinetic  energy 
and  thus  for  opposing,  by  the  pressure  of  these  mole- 
cules, the  very  cause,  namely  increased  external  pres- 
sure, which  induced  the  effect.1  An  increase  in  molecu- 
lar pressure  thus  occurs  to  meet  the  increase  in  the 
externally  applied  pressure. 

Suppose,  however,  that  as  fast  as  energy  is  liberated 
by  the  condensing  molecules  of  the  vapor  it  is  with- 
drawn from  the  system.  That  is,  suppose  the  change 
in  pressure  is  made  isothermally.  The  slightest  in- 
crease in  pressure  results  in  the  condensation  of  a  part 
of  the  vapor.  The  pressure  exerted  by  the  vapor  does 
not  rise  because  the  energy  released  by  condensation 
is  immediately  subtracted.  Condensation,  therefore, 
occurs  at  constant  pressure  and  temperature  until  all 
the  vapor  is  condensed.  This  phenomenon  is  usually 
shown  in  the  laboratory  by  using  a  barometer  tube, 
I,  as  in  Fig.  26.  A  drop  of  liquid  is  released  into  the 
vacuum  at  the  top  by  inserting  the  point  of  a  medicine 

1  This  reminds  us  of  "action  and  reaction."  Later  we  shall  dis- 
cuss the  principle  of  Le  Chatelier  which  has  to  do  with  such  cases. 

Q 


226 


THE   REALITIES  OF   MODERN   SCIENCE 


II 


FIG.  26. 


dropper  into  the  mercury  immediately  under  the 
column.  Evaporation  then  takes  place  from  this 
drop,  which  should  be  some  highly  volatile  substance, 
like  ether,  so  that  the  effect  will  be  marked  at  ordinary 
temperatures.  The  evaporation  continues  until  the 
number  of  molecules  in  the  space  above  the  liquid  is 
such  that  at  each  instant  just  as  many 
are  condensing  into  the  liquid  form  as 
there  are  evaporating  from  the  surface. 
A  statistical  equilibrium  is  thus  estab- 
lished between  the  liquid  and  its  vapor. 
The  pressure  exerted  by  the  vapor 
molecules  plus  that  of  the  weight  of 
the  column  of  mercury  (the  weight  of 
the  drop  of  liquid  is  negligible)  must 
balance  the  external  atmosphere.  The 
mercury  in'  the  column  therefore  falls.  The  amount 
that  this  column  is  reduced,  as  shown  by  a  second 
barometer  column,  II,  is  the  measure  of  the  vapor 
pressure  in  centimeters  of  mercury. 

If  the  barometer  tube  is  pushed  down  into  the  mer- 
cury the  liquid  condenses.  If  the  tube  is  moved  slowly 
the  temperature  will  remain  constant,  for  the  tube  and 
its  contents  will  lose  energy  to  their  surroundings  as 
fast  as  it  is  released  by  the  condensation.  The  reduc- 
tion of  the  volume,  occupied  by  the  liquid  and  its  vapor, 
will  be  found  to  occur  at  constant  pressure,  as  will  be 
evidenced  by  the  constancy  of  the  height  of  the  mercury 
volume.  Similarly,  if  the  volume  is  increased  by 
raising  the  tube  it  will  be  found  that  this  also  occasions 
no  change  in  the  pressure  as  long  as  any  liquid  is  in 
contact  with  the  vapor.  After  all  the  liquid  is  evap- 


CORRESPONDENCE  OF   MOLECULAR  STATES      227 


orated  the  mercury  column  will  rise  as  the  tube  is 
raised,  indicating  a  reduction  in  the  pressure  of  the 
vapor  with  increased  volume.  The  volume  and  pres- 
sure will  then  be  related  practically  as  for  a  perfect  gas. 

Let  us  now  make  a  plot  to  show  these  relations  of  P 
and  V  for  a  substance  which  is  maintained  at  a  tem- 
perature below  its  critical  temperature,  so  that  the 
liquid  and  its  vapor  may  coexist  in  equilibrium.  Let 
us  start  with  the  substance  entirely  in  the  form  of 
vapor.  Let  this  condition  be  represented  by  the 
point  a  of  Fig.  27.  If  x 

the  pressure  is  increased  * 
the    volume    decreases 
practically  inversely, 
and  successive  states  are 
represented    by    points 
along     the     hyperbolic 
curve  ab.    At  6  the  pres- 
sure is  such  that  there  Vl  Va  v 
is    some    condensation. 

This  pressure  is  the  vapor  pressure  of  boiling  for  the 
temperature  of  this  isothermal.  As  we  have  just  seen, 
this  pressure  cannot  be  increased  as  long  as  the  sub- 
stance is  maintained  at  the  same  temperature.  An 
attempt  to  increase  the  externally  applied  pressure 
merely  results  in  a  decrease  in  volume.  The  successive 
states  of  the  substance  are  now  represented  by  the 
solid  line  bf.  When  the  vapor  is  all  condensed  the 
volume  (represented  at  /)  is  merely  that  of  the  liquid. 
Further  increase  in  the  pressure  compresses  the  liquid, 
as  represented  by  the  line  fg. 

These  changes  may  be  carried  out  in  the  reverse 


228         THE   REALITIES  OF   MODERN  SCIENCE 

order,  by  starting  with  the  liquid  in  the  state  cor- 
responding to  g,  and  following  the  isothermal  in  the 
order  gfba  by  gradually  reducing  the  pressure.  In- 
cidentally, it  may  be  pointed  out  that  in  this  reversible 
cyclic  process  such  work  as  we  did  in  compressing  the 
substance  is  returned  to  us  as  it  expands.  Such 
energy  as  was  imparted  to  the  surroundings  during 
compression  is  also  returned  during  the  expansion. 

It  does  not  always  happen  that  the  change  follows 
exactly  this  isothermal,  abfg.  For  example,  it  is  pos- 
sible with  a  pure  liquid,  free  from  dissolved  air,  in  a 
vessel  with  smooth  walls,  to  obtain  the  following 
phenomenon.  Starting  from  the  state  indicated  by 
g  the  pressure  is  reduced.  When  /  is  reached  where 
boiling  should  occur,  it  may  not  occur.  The  pressure 
may  be  reduced  well  below  this  value  without  ebullition 
occurring.  The  V-P  plot  then  follows  the  partially 
dotted  line  gfe.  Suddenly,  however,  ebullition  occurs. 

A  somewhat  similar  phenomenon  may  occur  if  we 
start  with  the  substance  in  the  gaseous  condition,  as 
represented  at  a.  As  the  pressure  is  increased  the 
volume  is  decreased.  This  decrease  may  extend  along 
the  dotted  line  abc  well  beyond  the  point  b.  The  mole- 
cules are  thus  compressed  without  aggregating  into 
drops  of  liquid.  Suddenly  condensation  takes  place 
and  the  pressure  drops  to  the  boiling  point  pressure 
corresponding  to  the  temperature  of  the  isothermal. 
The  state  corresponding  to  b  is  called  that  of  saturated 
vapor.  From  a  to  b  the  vapor  is  below  saturation, 
that  is,  the  molecular  density  of  the  vapor  is  less  than 
that  of  a  vapor  at  the  same  temperature  which  is  in 
contact  with  its  liquid. 


CORRESPONDENCE  OF   MOLECULAR  STATES      229 

Saturated  vapor  is  in  equilibrium  with  its  liquid. 
Unsaturated  vapor  is  not,  for  it  tends  to  increase  in 
number  of  molecules  at  the  expense  of  the  liquid.  On 
the  other  hand,  supersaturated  vapor,  such  as  cor- 
responds to  states  represented  by  the  curve  6c,  is  in 
a  decidedly  unstable  equilibrium.1  The  slightest  cause 
will  result  in  a  disproportionate  effect.  Such  a  cause 
may  be  furnished  by  inserting  a  drop  of  the  liquid  or 
by  particles  of  dust  and  other  impurities. 

It  has  proved  impossible  as  yet  to  penetrate  very 
far  into  the  region  of  instability,  whether  above  b  or 
below  /.  It  is  usual  therefore  to  follow  the  suggestion 
of  James  Thomson  and  draw  a  dotted  line  like  cde  to 
represent  for  an  absolutely  pure  substance  the  part  of 
the  isothermal  which  we  cannot  follow  experimentally. 
Somewhere  between  6  and  /  the  substance,  which  is  a 
gas  at  b  and  liquid  at  /,  must  change  from  one  to  the 
other.  It  seems  reasonable  to  assume  that  such  a 
change  would  occur  continuously  in  some  such  way  as 
that  represented  by  cde.  Thus  the  pressure  rising 
above  b  might  ultimately  reach  a  maximum  at  c.  As 
it  decreases  it  might  well  fall  too  low,  reaching  a  mini- 
mum at  e.  Let  us  remember  that  the  process  of  aggre- 
gation into  a  liquid  is  a  contraction  of  the  substance, 
that  is,  a  motion  of  the  molecules  toward  each  other. 
Since  they  have  inertia,  or  mass,  such  a  motion  might 
well  result  in  an  excessive  contraction  corresponding 
to  the  state  e. 

In  the  states  represented  by  the  line  be  the  vapor  is 

1  Similar  phenomena  of  instability  of  state  occur  in  the  case  of 
liquids  cooled  below  the  freezing  point.  For  their  observation 
laboratory  conditions  are  usually  required. 


230         THE   REALITIES   OF    MODERN   SCIENCE 

compressed  and  has  an  excess  of  energy,  for  the  mole- 
cules have  not  yet  converted  their  p. e.  into  k.e.  The 
contraction  has  consumed  external  energy  instead  of 
available  internal  energy.  From  c  to  d  this  molecular 
p.e.  is  released.  When  it  is  converted  into  k.e.,  the 
molecules  may  travel  beyond  their  positions  of  stable 
equilibrium  until  their  relative  separations  are  too 
small,  as  represented  by  the  point  e.  As  the  molecules 
return  to  a  separation  corresponding  to  stable  equi- 
librium the  pressure  increases,  as  represented  by  the 
line  ef.  The  molecules  thus  arrive  at  the  same  con- 
dition as  if  the  successive  states  had  been  as  repre- 
sented by  the  line  bf,  and  there  had  been  no  over- 
shooting of  the  mark. 

We  cannot  carry  a  substance  through  this  hypotheti- 
cal isothermal,  for  we  cannot  carry  it  as  far  as  either  the 
maximum  or  the  minimum.  Van  der  Waals's  equation, 
however,  indicates  such  a  form  for  the  isothermal  of  a 
pure  substance.  Thomson's  hypothesis  appears  con- 
ceivable and  is  supported  to  some  extent  by  the 
experimental  entry  into  these  regions  of  instability. 
The  effects  which  scientists  observe  in  pressure  and 
temperature  are,  however,  average  effects  of  very  large 
numbers  of  molecules.  If  we  could  only  observe  the 
interactions  of  individual  molecules  we  should  need  no 
hypothesis.  But  if  we  could  observe  and  direct  the 
actions  of  individual  molecules  we  could  also  make 
possible  far  more  efficient  utilization  of  molecular 
energy  and  even  be  able  to  alter  the  processes  of  life 
itself.  It  is  the  intuition  of  scientists,  however,  that 
it  is  impossible  to  control  the  action  of  molecules  except 
in  so  far  as  their  average  effects  are  concerned. 


CORRESPONDENCE   OF   MOLECULAR  STATES      231 


FIG.  27. 


Returning  to  the  hypothetical  isothermal  of  Fig. 
27,  we  notice  that  for  any  pressure  higher  than  that 
corresponding  to  the  state  e  and  lower  than  that  of  c 
there  are  three  possible  volumes,  e.g.  those  marked 
Vi,  02,  and  v3  in  the  figure.  In  this  figure  are  also  plotted 
isothermals  for  higher  temperatures.  Consider  for 
example  that  lettered 
a'Vc'd'e'f'g'.  At  this 
higher  temperature  the 
molecules  occupying  a 
volume  corresponding 
to  a  are  at  the  higher 
pressure  corresponding 
to  a'.  Because  the  tem- 
perature is  higher,  that 
is  because  of  greater  k.e., 
the  external  pressure  which  must  supplement  the 
attractions  due  to  molecular  potential  energy  in  order 
to  cause  condensation  is  greater.  This  is  merely  another 
statement  of  the  fact  that  the  boiling  point  increases 
with  pressure.  The  volume  of  the  liquid  is  now  greater 
than  at  the  lower  temperature  because  liquids  expand 
if  the  molecular  k.e.  is  increased. 

The  distance  b'f  is  therefore  less  than  bf,  and  the 
dotted  portions  of  the  isothermal  should  be  less  pro- 
nounced. The  range  of  pressures  for  which  three 
volumes  are  possible  is  also  reduced.  The  maximum 
volume  occupied  by  the  liquid,  corresponding  to  /, 
increases  as  the  temperature  rises,  while  the  minimum 
volume  which  can  be  occupied  by  the  molecules  in  a 
stable  gas  condition,  corresponding  to  6,  decreases 
with  the  increase  in  temperature.  (For  some  tempera- 


232         THE   REALITIES   OF   MODERN  SCIENCE 

ture  these  two  volumes  are  the  same.  For  higher 
temperatures  the  substance  cannot  exist  as  a  liquid 
and  the  isothermals  are  the  hyperbolic  curves  of  a  gas.) 

As  the  temperature  rises  the  three  volumes  which  Van 
der  Waals's  equation  indicates  (two  of  which  exist  only 
on  the  hypothetical  portion  of  the  isothermal)  approach 
each  other  in  value.  For  some  particular  values,  say 
Pc  and  Tc,  they  will  be  equal.  This  occurs  when  the 
maximum  volume  of  the  liquid  and  the  minimum 
volume  of  the  saturated  vapor  are  equal.  This  is 
the  critical  volume,  and  Pc  and  Tc  are  the  critical  pres- 
sure and  temperature. 

The  equation  of  Van  der  Waals  has  value  as  indicat- 
ing the  general  character  of  the  relations  between  the 
molecules  of  any  substance  and  the  fact  that  all  sub- 
stances behave  essentially  alike,  but  it  gives  only 
approximate  relations.  It  fails  particularly  to  give 
accurate  values  for  the  critical  state. 

It  has  been  found,  however,  that  an  extension  of  this 
equation  may  be  used  to  considerable  advantage. 
Instead  of  using  P,  V,  and  T  to  represent  the  actual 
pressures,  volumes,  and  temperatures,  there  are  sub- 
stituted the  fractions  which  these  are  of  the  critical 
values.  For  example,  when  two  different  substances 
are  under  pressures  which  are  the  same  fractions  of 
their  respective  critical  pressures,  they  are  said  to  be 
in  corresponding  states  of  pressure.  Similarly  there 
are  corresponding  temperatures  and  volumes.  These 
substitutions  give  rise  to  an  equation  which  should  be 
applicable  to  all  substances.  All  substances,  since 
they  are  composed  of  molecules,  provided  the  mole- 
cules do  not  change  (chemically),  should  act  in  the 


CORRESPONDENCE  OF   MOLECULAR  STATES     233 

same  way,  and  pass  through  "  corresponding  states." 
Experiments  and  calculations  have  been  made  for  a 
large  number  of  substances  to  determine  whether  or 
not  this  is  so,  and  it  has  been  found  very  nearly  true. 
The  differences  are  small  for  substances  which  are 
somewhat  similar  chemically,  but  greater  for  other 
substances. 


CHAPTER  XVII 

MOLECULAR  MIXTURES 

THE  law  for  the  pressure  l  of  a  mixture  of  two  or 
more  gases  resulted  from  experiments  by  Dalton  which 
originated  in  his  desire  to  explain  the  mixture  of  gases 
in  which  men  were  just  realizing  that  they  were  living. 
In  our  study,  some  120  years  later,  we  may  start  from 
the  concept  of  a  molecular  and  atomic  composition 
and  thus  obtain  general  expressions  for  the  molecular 
condition  of  homogeneous  substances.  In  this  chapter 
we  shall  consider  mixtures  of  molecules. 

In  the  case  of  a  mixture  of  a  gas  and  a  liquid  it  is 
usual  to  say  that  the  gas  is  dissolved  by  the  liquid. 
Consider  the  case  where  there  is  no  chemical  action 
between  the  two  kinds  of  molecules.  Let  the  liquid 
be  contained  in  a  cylinder  which  we  then  fill  with  gas 
and  compress  by  a  piston.  Above  the  liquid  surface 
there  is  a  mixture  of  two  aeriform  substances,  namely 
the  gas  and  the  vapor  of  the  liquid,  and  below  the 
surface  the  mixture  of  liquid  and  gas  which  we  call  a 
solution.2  The  pressure  on  the  piston  is  the  sum  of 

1  Dalton's  law  is  obviously  only  an  approximate  relation.     Chem- 
ically inert  gases,  provided  their  volumes  are  well  above  the  critical 
volumes,  act  like  perfect  gases,  in  that  the  total  pressure  due  to 
molecular  impacts  is  the  sum  of  the  partial  pressures  which  the 
molecules  of  each  gas  would  exert. 

2  There  is  no  hard  and  fast  line  between  a  mixture  and  a  solution 
when  the  composition  of  the  molecules  is  not  altered.     The  substance 
of  smaller  amount  is  usually  said  to  be  dissolved  in  that  of  larger 
amount,  —  the  solute  in  the  solvent. 

234 


MOLECULAR   MIXTURES  235 

those  of  the  vapor  and  the  gas.  As  the  piston  is  moved 
into  the  cylinder  the  gas  pressure  rises.  (The  vapor 
pressure  does  not  change  at  constant  temperature.) 
Some  of  the  gas  molecules  are  thus  forced  into  the 
liquid.  At  any  instant  and  for  any  temperature  there 
will  be  a  definite  number  of  them  in  each  c.  c.,  and  this 
amount  of  dissolved  gas  is  proportional  to  the  "  partial 
pressure"  of  the  gas. 

The  surface  of  the  liquid,  which  looks  so  smooth 
and  continuous,  is  formed  by  moving  molecules.  There 
are  always  spaces  through  which  a  gas  molecule  may 
penetrate.  The  volume  thus  formed  by  the  inter- 
spaces of  the  liquid  molecules  communicates  at  the 
surface  with  the  volume  occupied  by  the  "undis- 
solved"  molecules  of  the  gas.  On  the  average  the 
effect  is  that  of  two  communicating  vessels  for  the 
gas,  one  free  from  obstructions  and  the  other  much 
reduced  in  actual  volume  because  of  obstructions. 
That  the  latter  are  moving  makes  no  difference  on  the 
average.  The  pressures  in  the  two  vessels  are  equal, 
for  otherwise  more  molecules  would  pass  from  one 
vessel  than  enter  it  from  the  other.  Since  the  pres- 
sures are  the  same,  the  number  of  molecules  per  c.  c.  of 
free  space  will  be  the  same  and  will  increase  directly 
with  an  increase  in  pressure. 

According  to  this  picture  the  molecules  of  the  dis- 
solved gas  behave  exactly  like  the  gas  molecules  which 
we  discussed  in  arriving  at  the  equation  PV  =  RT. 
In  applying  this  it  is  usual  to  measure  the  volume  in 
liters  (1000  c.  c.)  instead  of  c.  c.  and  hence  the  value  of 
R  must  be  1/1000  of  its  former  value.  We  are  of 
course  dealing  with  1  mole  of  the  gas,  and  hence  if  the 


236         THE   REALITIES   OF    MODERN   SCIENCE 

volume  is  1  liter  the  "  concentration "  of  gas  molecules 
is  1  mole  per  liter.  In  general  if  the  volume  is  F,  the 
concentration,  represented  by  c,  is  l/V.  In  the  equa- 
tion PV  =  RT,  if  we  substitute  c  for  l/V  we  have 
P  =  RTc,  as  the  expression  for  the  pressure  of  the  dis- 
solved molecules  of  the  gas. 

This  equation  is  not  limited  to  the  case  we  have  dis- 
cussed. No  matter  how  the  molecules  are  introduced 
into  the  liquid  we  should  expect  this  pressure.  For 
example,  the  molecular  weight  of  cane  sugar,  which  has 
the  formula  C12H22On,  is  12(12)  +  22(1.008)  +  11(16) 
or  342.2.  Ten  grams  of  sugar  in  a  liter  of  water  is 
10/342  of  a  mole  per  liter.  The  value  of  c  is  then 
0.029.  Suppose  the  solution  is  at  30°  C.  or  approximately 
300°  absolute.  If  we  calculate  the  pressure  which  the 
molecules  of  the  sugar  would  exert  on  each  sq.  cm.  of 
a  membrane  inserted  in  the  solution,  we  find  it  to  be 
about  0.7  of  an  atmosphere  or  10  pounds  to  the  square 
inch. 

In  the  case  of  a  gas  we  can  measure  such  a  pressure 
directly  by  measuring  that  of  the  gas  above  the  solu- 
tion, but  it  is  neces- 
sary, if  we  wish  to  check 
these  assumptions  for 
the  sugar  solution,  that 
we  should  measure  the 

pressure  in  the  solution.  Imagine  now  a  vessel  like 
that  of  Fig.  28  to  be  divided  by  a  membrane  which 
separates  solutions  of  the  same  character  but  of 
different  concentrations.  There  will  be  a  pressure  of 
pi  =  RTci  on  the  left  of  the  membrane  and  of  pz  =  RTc2 
on  the  right.  Assume  that  the  membrane  is  movable 


MOLECULAR   MIXTURES  237 

as  a  whole  and  that  it  is  permeable  by  the  molecules  of 
the  solvent,  but  impermeable  to  those  of  the  solute. 
The  net  pressure  exerted  on  the  membrane  by  the  dis- 
solved molecules  is  of  value  p\  —  p2,  acting  from  left  to 
right  if  pi  is  greater  than  pz.  The  membranous  piston 
will  then  move  toward  the  right.  As  it  does,  mole- 
cules of  the  solvent  pass  through  it,  so  that  the 
number  on  the  left  increases  while  that  on  the  right 
decreases.  Since  the  number  of  molecules  of  the  solute 
on  each  side  of  the  piston  remains  fixed,  the  concentra- 
tion on  the  left  decreases  and  that  on  the  right  in- 
creases.1 This  will  cease  when  the  concentrations 
have  been  altered  to  such  an  extent  that  pi— p2  =  0, 
that  is,  when  Ci  =  c^. 

In  the  case  we  have  discussed  there  is  no  external 
opposition  to  the  motion  of  the  membrane,  which  will 
move  into  its  final  position  without  there  being  any 
work  done.  The  case  is  similar  to  that  of  an  expand- 
ing gas.  If  the  two  vessels  in  Joule's  experiment 
(cf.  page  219)  were  separated  by  a  frictionless  and 
weightless  piston  the  piston  would  have  moved  with- 
out any  work  being  done.  It  is  only  when  a  decrease 
in  the  concentration  of  the  gas  molecules  takes  place 
against  an  external  force  that  work  is  done.  Suppose, 
however,  that  we  turn  the  vessel  of  Fig.  28  on  end 
and  oppose  the  motion  of  the  piston  by  weights.  The 
piston  will  move  just  as  before,  supposing  that  the 
weights  are  constantly  reduced,  being  always  just  a 

1  It  is  customary  to  speak  of  this  pressure  which  the  molecules 
of  a  solute  exert  as  the  osmotic  pressure  of  the  solution.  If  the 
membrane  is  withdrawn  the  concentrations  would  be  made  alike 
by  diffusion.  In  other  words,  osmotic  pressure  is  the  agency  causing 
diffusion. 


238         THE   REALITIES  OF   MODERN  SCIENCE 

bit  less  l  than  the  value  which  pi—pz  may  have  cor- 
responding to  each  position.  Then  work  will  be  done 
by  the  system  composed  of  the  two  solutions.  Of 
course,  this  work  is  at  the  expense  of  the  molecular 
energy  of  the  system,  and  the  solutions  will  be  cooled. 

It  is  possible  to  obtain  a  semipermeable  membrane 
which  allows  water  molecules  to  pass  freely  but  re- 
tains the  solute.  Of  this  character  are  the  cell  walls 
of  plant  and  animal  organisms ;  in  life  the  phenomena  of 
osmosis  are  of  great  importance.  Organic  membranes 
are  not,  however,  suitable  for  a  laboratory  experiment 
on  osmotic  pressures,  but  a  convenient  one  may  be  ob- 
tained by  the  precipitate  of  copper  ferrocyanide  which 
results  from  the  reaction  of  copper  sulphate  and  potas- 
sium ferrocyanide.  Since  this  will  not  stand  large 
forces  it  is  usual  to  cause  the  precipitation  to  take  place 
in  the  pores  of  an  unglazed  earthenware  vessel  or  cell. 
In  this  way  a  number  of  small  membranes  are  obtained, 
the  effect  of  which  is  the  same  as  that  of  a  single  large 
surface. 

The  cell  is  partially  filled  with  the  solution  of  which 
we  wish  to  measure  the  osmotic  pressure  and  immersed 
in  a  vessel  of  pure  solvent.  In  other  words,  we  make 
the  difference  in  concentration  between  the  two  solu- 
tions equal  to  that  of  the  solution  actually  under  test 
by  making  one  of  them  pure  solvent.  Now  we  have 
seen  that  the  two  solutions  tend  to  come  to  the  same 
pressure  for  their  dissolved  molecules.  The  solvent, 
e.g.  water,  enters  the  cell,  making  the  solution  weaker 

1  This  "bit  less"  is  of  course  negligible,  for  it  need  only  be  enough 
to  give  a  negligible  acceleration  to  a  weightless,  that  is,  inertialess, 
piston. 


MOLECULAR   MIXTURES  239 

and,  of  course,  increasing  the  height  of  the  column. 
An  equilibrium  position  is  finally  reached  when,  for  a 
further  increase,  the  work  done  by  the  solutions  would 
be  less  than  the  corresponding  increase  in  gravitational 
potential  energy  of  the  column  of  liquid.  The  osmotic 
pressure  is  then  taken  as  equal  to  that  of  the  column 
of  liquid. 

In  a  paper  before  the  Swedish  Royal  Academy  in 
1885,  Van't  Hoff  advanced  the  explanation  of  osmosis 
as  due  to  the  kinetics  of  the  molecules  themselves.  He 
found  confirmation  of  this  theory  not  only  in  the 
experimental  facts  that  osmotic  pressure  does  vary 
directly  as  the  absolute  temperature  and  as  the  concen- 
tration of  the  solution,  in  other  words  in  the  same 
way  as  gaseous  pressure,  but  also  in  previous  experi- 
ments on  the  effect  of  dissolved  substances  in  lowering 
the  freezing  point  and  in  raising  the  boiling  point  of 
solutions. 

As  to  the  last  of  these  phenomena,  we  have  seen  from 
the  isothermals  of  page  231  that  for  any  pressure  there 
is    a    boiling-point   tem- 
perature, namely  that  at 
which  the  pressure  of  the 
saturated  vapor   of    the 
liquid  equals  the  external 
pressure.    The  relation  of 
boiling-point    and    pres- 
sure   is    illustrated    for  FIG  29 
water  in  Fig.  29.      Let 

us  now  see  how  this  temperature  is  altered  by  mixing 
with  the  liquid  some  other  substance.  The  case  of 
immediate  interest  to  us  in  connection  with  osmosis 


p 


240         THE  REALITIES  OF   MODERN  SCIENCE 

is  that  of  a  pure  liquid  (a  solvent)  and  a  solid  (the 
solute).  The  pressure  exerted  by  molecules  is  always 
proportional  to  their  number  per  unit  volume  and  to 
their  kinetic  energy.  Assume  N+n  molecules  per  c.  c. 
of  the  pure  solvent.  Let  the  pressure  of  its  saturated 
vapor  be  p,  then  p&N+n.  Now  suppose  n  molecules 
are  replaced  by  those  of  a  dissolved  solid,  and  are  in- 
capable of  existing  in  a  vapor  state  at  this  temperature. 
The  molecules  available  for  producing  the  vapor  pres- 
sure are  now  N  per  c.  c.  and  the  resulting  pressure 
Hence  p^p'&N+n—  N  and 


(1) 

This  equation  gives  the  fractional  reduction  in  vapor 
pressure  in  terms  of  the  molecules  per  c.  c.  of  the  pure 
solvent  and  of  the  solute.  The  reduction  is  seen  to  be 
independent  of  the  substance  which  is  dissolved,  pro- 
vided the  number  of  molecules  per  c.  c.  is  the  same.  If 
the  vapor  pressure  is  reduced  the  solution  cannot  boil 
against  a  given  external  pressure  at  the  same  tempera- 
ture as  would  the  pure  solvent,  and  so  its  temperature 
must  be  raised  1  above  the  normal  boiling  temperature 
for  that  pressure. 

The  molecules  of  the  solute  exert  an  osmotic  pres- 
sure, and  Van't  Hoff  showed  that  it  was  possible  to 
calculate  the  lowering  of  the  vapor  pressure  from  a 

1  This  phenomenon  has  only  recently  entered  quantitatively  into 
the  daily  life  of  the  home  in  the  matter  of  making  jellies  and  candies. 
While  water  will  boil  at  212°  F.  under  atmospheric  pressure,  the 
boiling  point  of  a  solution  of  sugar  is  higher,  depending  upon  its  con- 
centration. Since  other  substances  like  fruit  acids  also  enter  into 
the  solution,  boiling  temperatures  furnish  a  convenient  test  of  the 
concentration,  and  recipes  are  to-day  expressed  in  such  terms. 


MOLECULAR   MIXTURES 


241 


knowledge  of  this  pressure.  He  also  developed,  on 
the  basis  of  a  perfect  gas  relation  for  the  dissolved 
molecules  of  a  solution,  an  expression  for  osmotic  pres- 
sure in  terms  of  the  lowering  of  the  freezing  point. 
This  relation  need  not  be  derived,  but  the  physical 
phenomenon  illustrates  some  interesting  principles. 

Solid  substances,  like  liquids,  lose  molecules  as 
vapor  although  in  most  cases  the  vapor  pressure  x  is 
too  small  to  be  meas- 
ured. As  the  tempera- 
ture increases  the  vapor 
pressure  rises.  We  may 
plot  pressure  and  tem- 
perature for  a  solid  in 
equilibrium  with  its 
vapor,  just  as  in  Fig.  29 
of  page  239  we  plotted 
the  relation  for  the 

liquid  in  equilibrium  with  its  vapor.  These  two  curves 
will  intersect  as  at  0  in  Fig.  30.  Curve  A  is  for  solid- 
vapor  equilibrium  and  B  is  for  liquid- vapor  equilibrium. 
At  0  is  represented  the  condition  at  which  all  three 
states  may  coexist  in  equilibrium.  0  is  therefore  called 
a  "  triple  point." 

This  point  is  the  freezing  point  (or  the  melting  point, 
depending  upon  our  point  of  view).  Under  the  con- 
ditions it  represents,  the  vapor  arising  from  the  solid 
is  in  equilibrium  with  that  rising  from  the  liquid.  But 
suppose  that  the  vapor  pressure  of  the  solid  was  not 
just  equal  to  that  of  the  liquid.  Either  the  solid  or 

1  For  solid  benzine  at  5.5°  C.  the  pressure  is  3.55  cm.  of  mercury, 
and  for  ice  at  about  0°  C.  it  is  0.46  cm. 


242         THE   REALITIES   OF   MODERN   SCIENCE 

the  liquid  would  grow  at  the  expense  of  the  other. 
Now  this  is  exactly  what  does  happen  if  the  temperature 
is  altered.  Thus  you  notice  that  as  the  temperature 
falls  below  that  of  the  triple  point  the  pressure  of  the 
supercooled  liquid,  shown  by  the  dotted  extension  of 
the  curve  B,  is  higher  than  that  of  the  solid.  Molecules 
leaving  the  surface  of  the  water  as  vapor  then  serve  to 
supersaturate  the  vapor  above  the  solid  and  thus  con- 
dense on  it.  In  other  words  some  of  the  water  freezes. 
Conversely  it  may  be  seen  that  the  curve  A  is  steeper 
at  0  than  is  curve  1  B,  so  that  if  the  temperature  rises 
above  the  freezing  point  the  solid  loses  molecules  to 

its  vapor  more  rapidly 
than  it  gains  from  it 
and  there  is  thus  an  ex- 
cess  of  molecules  which 
condense  as  a  liquid. 
Melting  thus  occurs. 

If  a  substance  is  dis- 
solved in  the  liquid, 
for  which  we  have  the 


curves  in  Fig.  31,  its 

vapor  pressure  is  reduced  according  to  equation  (1). 
The  vapor  pressure  curve  B'  corresponding  to  a  solu- 
tion is  shown  dotted.  The  intersection  of  B',  with  A 

1  Curves  like  A  and  B  represent  equilibrium  conditions.  Thus  for 
any  temperature  greater  than  freezing  there  is  one  and  only  one 
pressure  at  which  the  liquid  and  its  vapor  may  coexist  in  equilibrium, 
and  that  is  the  pressure  corresponding  to  this  temperature  on  the 
curve.  We  may  use  the  term  "reaction"  to  describe  these  actions 
and  indicate  by  arrows  the  direction  in  which  a  reaction  occurs. 
Thus  for  water  we  have  H20  (liquid)^  H2O  (vapor).  The  double 
arrows  are  made  similar  to  indicate  that  this  reversible  reaction  is 
in  equilibrium. 


MOLECULAR   MIXTURES  243 

is  evidently  at  a  point  corresponding  to  a  lower  tem- 
perature, that  is,  the  freezing  point  of  the  solution  is 
less  than  that  for  the  pure  solvent. 

Let  us  now  summarize  as  to  molecular  pressures  in 
liquids.  The  lowering  of  the  freezing  temperature  and 
the  raising  of  the  boiling  temperature  are  both  the  re- 
sult of  the  reduction  in  the  vapor  pressure  of  a  liquid 
which  occurs  when  some  of  the  molecules  are  not  avail- 
able for  producing  vapor.  Within  the  liquid  those 
molecules  which  are  not  available  for  producing  vapor 
exert  a  partial  pressure  known  as  the  osmotic  pressure 
of  the  solution.  These  phenomena  depend  upon  the 
number  of  such  molecules  per  unit  of  volume. 

It  is  however  more  accurate  to  say  that  they  depend 
upon  particles  of  essentially  molecular  size.  This 
allows  for  cases  where  the  number  of  particles  entering 
into  the  reaction  differs  from  the  number  of  molecules 
of  the  substances  involved  either  because  of  a  disso- 
ciation or  of  a  polymerization  of  the  molecules.  Van't 
Hoff's  equation,  therefore,  is  usually  written  as 
P  =  RTic,  where  the  factor  i  is  unity  for  those  cases 
like  sugar  solutions  where  there  is  a  normal  molecular 
condition. 


CHAPTER  XVIII 

ELECTROLYTIC  DISSOCIATION 

THE  earliest  indications  of  the  electrical  character 
of  matter  were  obtained  from  studies  of  the  conduction 
of  electricity  through  solutions.  To-day  we  know  that 
such  conduction  is  essentially  a  mechanical  process  of 
transferring  electrons.  In  order  that  a  medium  which 
has  no  dislodged  electrons  shall  serve  to  transfer 
through  itself  electricity,  there  must  be  brought  about 
and  maintained  an  abnormal  electronic  condition  for 
some  of  its  molecules.  Normal  molecules  will  not  so 
serve. 

This  abnormality  can  only  occur  if  the  molecule  is 
dissociated  into  parts  which  are  abnormal.  If  there 
are  thus  formed  two  components  they  must  depart 
equally  but  oppositely  from  normality,  that  is,  they 
must  be  equally  and  oppositely  charged.  Even  then 
conduction  can  occur  through  such  a  medium  only  if 
the  abnormal  particles  are  free  to  move,  as  would  be 
the  case  if  the  medium  were  a  liquid.  Now  pure 
liquids  like  water,  alcohol,  or  hydrochloric  acid  are 
non-conductors.  Solutions  of  substances  which  are 
chemical  compounds,  frequently  but  not  generally, 
are  conductors.  Those  which  conduct  are  called 
electrolytes  or  sometimes  "ionogens." 

The  simplest  dissociation  would  be  that  in  which  a 

244 


ELECTROLYTIC   DISSOCIATION  245 

molecule  formed  two  "ions,"  one,  the  anion,  being 
abnormal  by  an  additional  electron,  and  the  other,  the 
cation,  abnormal  by  a  corresponding  deficiency.  The 
anion  of  an  electrolyte  is  not  a  discrete  electron  as  in 
metals,  nor  is  it  formed  by  the  addition  of  an  electron 
to  one  of  the  original  normal  molecules  as  is  sometimes 
the  case  in  gases.  In  electrolysis  both  ions  are  mo- 
lecular (or  atomic)  in  size  and  in  composition  except 
for  the  abnormality  in  number  of  electrons.  More 
than  two  ions  may  be  formed  from  complex  molecules, 
but  only  in  a  few  of  the  simplest  cases  is  the  number 
equal  to  the  number  of  atoms  in  the  molecule.  In 
general  it  is  less  than  the  number  of  atoms  composing 
the  normal  molecule. 

Consider,  for  example,  hydrochloric  acid.  The  pure 
acid  and  pure  water  are  non-conducting,  but  a  mixture 
of  the  two  is  conducting.  This  may  be  observed  in  a 
simple  manner  by  inserting  two  chemically  inert  elec- 
trodes, e.g.  platinum,  and  connecting  a  battery  and  a 
current-measuring  instrument  to  the  electrodes.  Hy- 
drogen gas  will  rise  from  the  electrode  connected  to 
the  negative  terminal  of  the  battery,  that  is,  from  the 
cathode.  Chlorine  gas  will  rise  from  the  anode.  We 
therefore  conclude  that  the  dissociation  which  has  made 
conduction  possible  is  that  of  the  HC1  rather  than  of 
the  H2O  of  the  solution.  The  gases  which  rise  are  the 
diatomic  gases  H2  and  C12  and  are  composed  of  normal 
molecules. 

This  simple  experiment  raises  four  questions :  (1) 
What  causes  the  dissociation  which  is  evidenced  by  the 
conductivity  of  the  solution?  (2)  How  do  the  anion 
and  the  cation  neutralize  their  respective  abnormal- 


246         THE   REALITIES   OF    MODERN   SCIENCE 

ities,  that  is,  how  do  they  deliver  up  their  charges? 
(3)  Why  are  these  gases  diatomic?  and  (4)  Why  do 
normal  molecules  instead  of  abnormal  molecules  com- 
pose the  liberated  gases?  There  is  no  question  as  to 
the  mechanism  of  the  transfer  of  the  ions  through  the 
liquid,  for  that  is  the  result  merely  of  superimposing  on 
their  normal  haphazard  motions  a  directed  motion  due 
to  the  potential  of  the  electrodes.  The  answers  to 
these  questions  should  be  considered  as  indicative 
rather  than  conclusive,  for  there  are  many  unsettled 
matters  involved. 

(1)  The  normal  molecule  in  this  case  consists  of  two 
nuclei  and  their  associated  electrons.  The  two  nuclei 
are  not  combined  into  one  larger  nucleus,  but  are  the 
individual  nuclei  of  hydrogen  and  chlorine  atoms. 
About  neither  of  them,  however,  is  the  configuration 
of  electrons  what  it  would  be  about  isolated  atoms  of 
the  two  component  elements.  The  nuclei  pellate  and 
the  electrons  pellate,  but  any  electron  tractates  with 
either  of  the  two  nuclei.  We  are  not  yet  able  to  form 
a  geometrical  picture  of  such  a  system,  so  let  us  draw 
an  analogy  from  human  relations.  Let  us  consider  a 
political  coalition  formed  by  two  leaders  of  exceptional 
individuality  and  diverse  tendencies.  About  them 
are  a  group  of  adherents,  men  who  require  leadership 
and  are  held  in  the  coalition,  despite  their  petty  sus- 
picions of  each  other,  by  the  attraction  of  the  leaders. 
The  leaders,  on  the  other  hand,  are  kept  from  diverging 
by  the  group  about  them,  whose  adherence  is  to  leader- 
ship itself  rather  than  to  any  individual  leader.  Such 
a  coalition  may  dissolve  under  the  attraction  of  dis- 
similar groups  and  in  so  doing  form  two  new  groups. 


ELECTROLYTIC   DISSOCIATION  247 

The  adherents  do  not,  however,  necessarily  apportion 
themselves  to  the  individual  leaders  exactly  in  propor- 
tion to  their  relative  strengths.  In  one  of  these  new 
groups  the  adherents  are  in  excess  of  the  leader's 
capacity  while  in  the  other  they  are  too  few.  The 
dissociated  groups  may  later  be  thrown  together  and 
recombine.  Similarly  the  dissociation  of  a  molecule 
in  a  solution  is  the  result  of  the  formation  of  new 
systems  of  potential  energy  between  the  molecules  of 
the  solvent  and  the  component  ions  of  the  solute 
molecules. 

(2)  The  battery  or  other  source  of  potential  differ- 
ence draws  electrons  from  one  platinum  electrode,  the 
anode,  and  forces  electrons  to  the  other,  the  cathode. 
The  cation,  or  positive  ion,  on  reaching  the  cathode 
receives  from  it  an  electron,  thus  becoming  a  normal 
molecular    particle.     The    neutralization    of    its    de- 
ficiency in  electrons  corresponds  exactly  to  the  neu- 
tralization of  the  positive  charge  of  a  pith  ball  when  it 
is  attracted  to  and  touches  a  negatively  charged  ebon- 
ite rod.     Similarly,  the  anion  gives  up  an  electron  to 
the  anode.     The  number  of  electrons  in  the  solution  is 
then  unaltered  on  the  average,  and  similarly  as  to  the 
number  in  the  part  of  the  conducting  circuit  formed  by 
the  battery  and  its  connecting  wires. 

(3)  The  electrolytic  ions  are  not  the  particles  into 
which  the  substance  of  the  solute  would  decompose  as 
a  gas.     For  example,  NH4C1  decomposes  as  a  gas  into 
ammonia    (NH3)    and   hydrochloric    acid    (HC1).     In 
solution  the  dissociation  is  into  an  ion  formed  by  the 
atoms  NH4,  which  is  deficient  by  one  electron,  and  an 
ion  formed  by  a  chlorine  atom  and  one  extra  electron* 


248         THE   REALITIES  OF   MODERN  SCIENCE 

These  ions  do  not  appear  as  vapor,  for  their  existence  is 
due  to  the  potential  energy  of  the  systems  which  they 
form  with  the  molecules  of  the  solvent.  The  kinetic 
energy  which  would  be  required  to  move  one  of  them 
beyond  the  influence  of  the  molecules  of  the  solvent  is 
greater  than  that  corresponding  to  the  energy  required 
to  produce  an  evaporation  of  the  solvent  molecules.1 

(4)  At  the  instant  that  the  ions  neutralize,  normal 
electronic  groups  are  formed,  atoms  in  this  case  of 
hydrochloric  acid.  Such  neutralization  means  a  new 
configuration  of  electrons.  The  hydrogen  atom  is 
therefore  in  a  condition  of  changing  electronic  con- 
figuration which  renders  it  peculiarly  adaptable  to  the 
formation  of  larger  aggregates.  The  aggregate  formed 
by  two  such  nascent  hydrogen  atoms  appears  to  be 
very  stable.  Diatomic  molecules  are  the  product  of 
the  neutralization.  These  are  the  H2  molecules  which 
we  know  as  hydrogen  gas.  A  similar  formation  of  C12 
occurs  at  the  other  electrode.  These  gases  are  not 
especially  soluble  at  ordinary  pressures  and  temper- 
atures, and  therefore  they  rise  as  gases  above  the 
liquid  surface. 

That  there  is  an  electrolytic  dissociation  we  have 
reasoned  from  our  knowledge  of  the  mechanism  of  the 
conduction  of  electricity.  Historically,  however,  the 
proof  of  the  dissociation  was  largely  the  result  of 

1  The  latter  evaporate  rather  than  the  ions  of  the  solute.  If 
the  temperature  is  increased  the  solvent  molecules  evaporate  more 
rapidly,  and  after  a  time  the  concentration  of  the  solute  ions  be- 
comes such  that  normal  molecules  of  the  solute  are  formed  in  greater 
numbers  each  instant  than  they  are  dissociated.  The  result  is  a 
decrease  in  the  total  average  number  of  ions  in  the  solution,  until 
finally  when  all  the  solvent  is  evaporated  the  original  solute  is  left. 


ELECTROLYTIC   DISSOCIATION  249 

measurements  upon  the  osmotic  pressure,  the  freezing 
point,  and  the  boiling  point  of  such  solutions. 

The  amount  of  the  dissociation  may  be  conveniently 
calculated  from  such  observations.  Suppose  that  there 
are  N  molecules  of  solute  and  that  a  fraction  a  of  these 
are  rendered  abnormal  by  dissociation.  Then  there 
are  (l—a)N  normal  molecules  and  aN  dissociated 
molecules  of  the  solute.  Let  n  represent  the  number 
of  ions  into  which  each  dissociated  molecule  has  sepa- 
rated, then  the  total  of  molecular  particles  introduced 
into  the  solution  by  the  solute  is  not  N  but  is 

(l-a)N+naN=N(na-a+l)  =  N[(n-l)a+l]     (1) 

A  study  of  the  conductivity  of  an  electrolytic  solu- 
tion enables  us  to  determine  a  in  a  direct  and  very  con- 
venient manner.  If  the  two  electrodes  of  such  an  elec- 
trolytic cell  as  was  described  above  are  maintained  at 
a  constant  potential  difference  while  the  concentration 
in  moles  per  liter  of  the  solution  is  altered,  it  is  found 
that  the  current  varies,  indicating  a  variation  in  the 
ability  of  the  solution  to  conduct  electricity.  The 
conductivity  is  the  ratio  of  the  current  in  amperes  to 
the  potential  difference  in  volts.  A  solution  would 
then  have  unit  conductivity  if  it  carried  electricity  at 
the  rate  of  one  ampere  l  under  a  potential  difference 
of  one  volt. 

For  any  given  electrolyte  the  conductivity  will  de- 
pend upon  the  dissociation  a.  At  great  dilution  we  may 
consider  a  to  be  unity.  Or,  as  it  is  usually  expressed, 
the  dissociation  at  infinite  dilution  is  complete.  Under 
these  conditions,  the  ions,  which  are  formed  by  the  ac- 
1  One  coulomb  per  second.  Cf.  p.  178. 


250         THE  REALITIES  OF   MODERN  SCIENCE 

tion  of  the  solvent,  in  pursuing  their  haphazard  motions 
through  the  liquid  have  but  a  negligible  chance  of  col- 
liding with  ions  of  the  opposite  electrical  sign.  On  the 
other  hand,  if  the  dilution  is  not  infinite  there  will 
be  collisions  between  ions  of  opposite  sign,  and  hence 
at  any  instant  a  number  of  recombinations  are  taking 
place.  Of  course,  a  similar  number  of  molecules  are 
dissociating  at  this  instant  or  otherwise  the  solution 
would  be  changing  in  its  ionization.  The  number  of 
molecules  of  the  solute  is  increased  if  the  concentration 
is  increased.  Unless  therefore  the  dissociation  a  de- 
creases proportionately,  the  total  number  of  ions 
available  for  transferring  electricity  will  also  increase. 
There  will  be  a  higher  conductivity,  as  concentration 
increases,  until  a  point  is  reached  where  the  dissociation 
decreases  equally.  Then,  for  further  increases  in  con- 
centration the  dissociation  decreases  even  more  rapidly, 
resulting  finally  in  a  solution  of  essentially  zero  dis- 
sociation and  hence  negligible  conductivity. 

The  value  of  a  may  be  determined  from  measure- 
ments of  the  conductivity,  or  more  strictly  the  molecu- 
lar conductivity,  which  is  the  ratio  of  the  conductivity 
to  the  concentration.  The  molecular  conductivity  is, 
then,  an  actual  conductivity  of  the  solute  molecules 
and  changes  only  as  these  molecules  change  by  dis- 
sociation. Represent  it  by  L  and  let  K  be  the  factor  of 
proportionality.  Then  L  =  Ka.  Calling  the  maximum 
value  Loo  corresponding  to  the  dissociation  o=l, 
we  have  K  =  Loo.  Hence  the  dissociation,  a,  corre- 
sponding to  any  observed  molecular  conductivity, 
L,  is 

(2) 


ELECTROLYTIC   DISSOCIATION  251 

The  value  of  n,  the  number  of  ions  into  which  a 
molecule  of  the  solute  dissociates,  is  obtained  from  a 
study  by  weight  of  the  substances  liberated  as  gases  or 
deposited  on  the  electrodes,  when  different  solutions 
transfer  equal  quantities  of  electricity.  In  the  case 
of  hydrochloric  acid,  a  molecule  dissociates  into  one 
anion  and  one  cation  and  n  =  2.  Each  of  these  ions 
serves  to  transfer  one  electron.  Whatever  the  division 
may  be,  the  quantity  of  electricity  transferred  by  each 
ion  must  be  an  integral  number  of  electrons. 

Other  types  of  dissociation  are  possible  giving  higher 
values  of  n.  Thus  n  may  be  3,  as  is  the  case  for  barium 

hydroxide  Ba(OH)2  which  separates  into  the  cation, 

++ 

Ba,  deficient  by  two  electrons,  and  two  anions  of  the 

composition,  OH,  each  with  a  single  excess  electron. 
The  converse  may  also  happen ;  for  example  sulphuric 

acid,  H2S04,  separates  intojbwo  cations  of  H  and  a 
single  anion  of  the  form  SO4  which  carries  two  extra 
electrons. 

In  the  case  of  those  elements  which  have  more  than 
one  valence,  ions  may  be  formed  differing  in  their 
number  of  electrons,  depending  upon  the  compound 
from  which  they  are  derived.  Thus  iron  may  be 

divalent  and  form  the  ferrous  chloride  FeCl2,  in  which 

++  _ 

case  the  ions  are  Fe  and  two  Cl  ions.     On  the  other 

hand,  in  ferric  chloride  FeCl3  the  iron  atom  is  trivalent, 

and  this  molecule  in  solution  splits  into  an  iron  ion 

+++ 

Fe,  which  is  deficient  by  three  electrons,  and   three 

ions  of  chloride,  Cl,  each  with  an  extra  electron.     For 
this  case  n=4. 
The  same  element  may  appear  in  one  substance  as 


252         THE   REALITIES   OF   MODERN  SCIENCE 

an  anion  and  in  other  substances  as  a  cation,  thus  in 
ferrocyanide  of  potassium,  K4Fe(CN)6,  there  are  formed 

four  positive  jipns  of  potassium,  K,  and  one  complex 
anion,  Fe(CN)6,  composed  of  an  iron  atom  and  six 
molecular  groups  called  cyanogen,  each  composed  of 
an  atom  of  carbon  and  one  of  nitrogen.  The  cyanogen 
ion  consists  of  thirteen  atoms  and  carries  four  extra 
electrons.  The  iron  atom  in  this  case  is  not  a  cation 
but  a  component  part  of  the  anion,  and  thus  does 
not  enter  into  the  characteristic  chemical  reactions 
which  distinguish  it  in  those  cases  where  it  is  a  cation. 

When  an  ion  is  neutralized  at  an  electrode,  there  are 
three  possible  effects ;  thus  it  may  be  liberated  as  a 
gas,  be  deposited  on  the  electrode,  or  enter  into  a  re- 
action with  the  solvent.  The  electrolysis  of  HC1,  of 
silver  nitrate,  AgN03,  and  the  decomposition  of  water 
are  examples  respectively. 

Consider  a  solution  of  silver  nitrate  between  elec- 
trodes of  silver.  When  the  silver  ions  are  neutralized 
at  the  cathode  they  are  deposited,  granules  of  pure 
silver  being  formed  on  the  surface  of  the  plate.  At 
the  anode  the  neutralization  of  the  radical  NO3  takes 

place  by  its  combination  with  a  silver  ion,  Ag,  which  is 
released  by  the  anode  itself.  Thus  silver  is  trans- 
ferred from  the  anode  to  the  cathode.  We  may  con- 
sider N03  to  react  with  the  silver  of  the  anode  to  form 
AgN03,  or  better  we  may  consider  that  the  silver  dis- 
solves as  silver  ions,  that  is,  atoms  deficient  by  one 
electron  each.  A  surplus  of  electrons  is  thus  left  on 
the  anode  just  as  if  the  nitrate  radicals  had  given  up 
to  it  their  excess  electrons. 


ELECTROLYTIC   DISSOCIATION  253 

This  excess  is  withdrawn  by  the  action  of  the  bat- 
tery. This  process  is  evidently  the  basis  of  electro- 
plating, whereby  silver  is  deposited  upon  a  baser  metal, 
which  is  used  as  the  cathode.  Similarly,  the  electroly- 
sis of  copper  sulphate  is  used  commercially  either  for 
copperplating  or  for  the  refining  of  copper  derived 
from  ores. 

In  the  third  case  the  products  which  appear  are  the 
results  of  secondary  reactions  at  the  electrodes.  In 
the  case  of  a  solution  of  sodium  hydroxide,  NaOH,  the 

ions  are  Na  and  OH.  Now,  metallic  sodium  enters 
into  a  vigorous  reaction  with  water,  forming  hydro- 
gen gas  and  sodium  hydroxide,  2Na+2H2O,  giving 
H2-f2NaOH.  In  conduction  through  sodium  hy- 
droxide, when  the  sodium  ion  has  neutralized  its  de- 
ficiency of  one  electron,  it  is  in  a  condition  to  enter 
into  this  reaction.  The  formation  of  electrically  neu- 
tral atoms  of  sodium  is  followed,  then,  immediately 
by  the  decomposition  of  water  and  the  liberation  of 
molecules  of  hydrogen  gas.  At  the  anode  the  neu- 
tralized hydroxyl  ions  (OH)  form  water  and  liberate 
oxygen,  thus  4(OH)=O2+2H2O.  It  was  this  second- 
ary reaction  which  in  the  earlier  days  of  the  investi- 
gation of  electrolysis  obscured  to  some  extent  the 
actual  process  and  resulted  in  the  idea  that  an  electric 
current  decomposed l  water.  The  more  nearly  free 

1  A  similar  secondary  reaction  is  obtained  if  a  dilute  acid  is  used, 
as  for  example  sulphuric  acid.  In  that  case  hydrogen  is  liberated 
at  the  cathode  in  the  primary  reaction.  At  the  anode  the  water  is 
decomposed  by  the  neutralized  sulphate  radical  and  oxygen  is 
liberated,  thus 

2H20+2S04=2H2S04+0, 


254         THE   REALITIES  OF    MODERN  SCIENCE 

the  water  can  be  made  of  impurities  the  more  nearly 
is  it  non-conducting,  and  the  less  it  is  decomposed. 

Whether  or  not  the  liberated  product  is  the  result 
of  the  primary  or  the  secondary  reaction  there  is  in 
the  product  of  electrolysis  one  univalent  atom  for  each 
electron  transferred  through  the  solution,  or  in  gen- 
eral a  number  of  atoms  equal  to  the  quotient  of  the 
number  of  electrons  which  have  been  transferred 
divided  by  the  valence  of  the  product.  The  masses 
of  the  products  produced  by  the  transfer  through  dif- 
ferent electrolytic  solutions  of  equal  numbers  of  elec- 
trons should  therefore  be  proportional  to  the  so-called 
"electrochemical  equivalents"  of  the  individual  prod- 
ucts, i.e.  to  their  molecular  weights  per  unit  of  valence. 

Now  these  conclusions,  although  not  expressed  in 
terms  of  electrons,  were  obtained  experimentally  by 
Faraday  in  1834  and  are  known  as  his  first  and  second 
laws  of  electrolysis.  To-day,  it  is  partly  a  convenience 
to  speak  of  them  by  this  name  and  partly  a  matter 
of  justly  honoring  the  name  of  one  of  the  most  fruit- 
ful investigators  in  the  early  development  of  electricity. 
The  conclusions  are,  however,  inevitable  as  soon  as 
we  accept  as  facts  the  hypotheses  of  atoms  and  elec- 
trons. 

The  disadvantage,  in  this  instance  and  in  other 
similar  cases,  which  is  attached  to  the  retention  of  the 
names  of  the  laws,  is  that  it  results  usually  in  their 
acceptance  by  the  beginner  in  science  as  empirical 
laws  rather  than  as  the  logical  consequence  of  the 
structure  of  matter.  In  treating  electrolysis  we  have 
in  this  chapter  deliberately  inverted  the  historical 
order  so  as  to  emphasize  the  essential  unity  of  ap- 


ELECTROLYTIC  DISSOCIATION  255 

parently  discrete  phenomena.  The  number  of  laws 
and  concrete  facts  which  the  student  of  science  needs 
to  remember  are  fewer  than  the  names  which  have 
become  associated  with  them  in  their  evolutionary  de- 
velopment. If  the  reader  obtains  a  correct  concept  of 
molecules,  atoms,  and  electrons,  many  important  laws 
will  appear  to  him  as  deducible. 

In  electrolysis  the  number  of  electrons  leaving  the 
solution  at  the  anode  is  equal  to  that  entering  at  the 
cathode.  In  conduction  through  an  electrolytic  solu- 
tion it  may  however  happen  that  the  anion  and  the 
cation  move  through  the  solution  at  different  speeds. 
The  result  will  be  a  different  concentration  of  anions 
near  the  anode  than  of  cations  at  the  other  electrode. 
This  is  evidenced  in  cases  like  that  of  copper  sulphate 
CuS04  by  a  paling  of  the  color  near  the  anode,  but  it 
may  always  be  found  by  a  chemical  analysis  of  samples 
of  the  liquid  taken  from  near  the  two  electrodes. 

In  a  dilute  solution,  where  dissociation  is  essentially 
complete,  we  should  expect  the  behavior  of  an  ion  to 
be  characteristic  and  not  dependent  upon  the  par- 
ticular substance  from  which  it  was  derived.  Thus 
all  hydrogen  ions  are  alike  whether  they  are  obtained 
from  HC1,  HNO3,  H2S04,  or  some  other  acid.  The 
molecular  conductivity  of  a  dilute  solution  will,  there- 
fore, depend  upon  the  individual  characteristics  of  its 
anion  and  its  cation.  Since  the  amount  of  electricity 
transferred  across  a  solution  is  the  sum  of  that  trans- 
ferred by  the  two  kinds  of  ions,  the  conductivity 
should  be  the  sum  of  the  conductivities  of  the  two 
kinds  of  ions.  These  conductivities  are  called  their 
"mobilities."  Now,  an  acid  is  an  electrolyte  which 


256         THE  REALITIES   OF    MODERN  SCIENCE 

gives  a  hydrogen  cation ;  a  base,  one  which  gives  an 
hydroxyl  anion  (OH)  ;  and  a  salt  is  an  electrolyte  which 
has  neither  of  these  ions.  The  mobility  of  the  hydro- 
gen ion  is  the  highest  of  all,  OH  is  next,  and  other  ions 
much  less.  In  these  individual  mobilities,  therefore, 
we  see  the  explanation  of  the  physical  fact  that  acids 
are  better  conductors  than  bases  and  the  latter  better 
than  salts. 

Chemists  are  most  generally  interested  in  acids  as 
sources  of  the  H  ions  which  cause  the  characteristic 
effects.  For  this  reason  they  are  accustomed  to  speak 
of  acids,  like  HC1,  which,  in  a  solution  of  one  tenth 
mole  per  liter,  dissociate  more  than  70  per  cent,  as 
" strong."  If  the  dissociation  of  a  decinormal  solution 
of  the  acid  lies  between  10  and  70  per  cent  it  falls  into 
a  second  group  of  strong  acids.  On  the  other  hand, 
acids  like  acetic,  which  in  similar  concentrations  have 
dissociations  of  1  to  10  per  cent,  are  "  weak, "  and  those 
like  carbonic  acid,  which  have  even  smaller  values, 
are  " feeble."  In  the  same  way  the  chemist  divides 
bases  into  two  groups ;  those  like  sodium  hydroxide 
with  large  dissociations  are  called  "  strong, "  while 
those  which,  in  a  decinormal  solution,  form  hydroxyl 
ions  from  less  than  two  per  cent  of  the  molecules  are 
weak  bases.  Of  the  latter  ammonium  hydroxide  is  an 
example.  Salts  may  be  formed  by  weak  acids  and 
strong  bases  and  vice  versa;  thus  sodium  acetate  is 
the  salt  of  a  weak  acid  and  a  strong  base. 


CHAPTER  XIX 
EQUILIBRIA  [AND  THEIR  DISPLACEMENT 

THE  idea  of  a  statistical  or  mobile  equilibrium  we 
have  met  in  considering  a  system  of  two  phases,  e.g.  a 
liquid  and  its  vapor.  When  the  system  is  in  equilib- 
rium there  are  just  as  many  molecules  of  the  liquid 
evaporating  from  the  surface  each  instant  as  there  are 
molecules  of  vapor  condensing  into  the  liquid.  If  the 
temperature  is  raised  evaporation  increases  until  the 
vapor  density  corresponds  to  the  new  temperature. 
As  the  density  of  vapor  molecules  increases  there  is  a 
corresponding  increase  in  the  number  of  them  which 
strike  back  into  the  liquid  surface.  When  the  rate  at 
which  molecules  condense  has  increased  sufficiently 
to  be  equal  to  the  rate  at  which  they  evaporate  a  new 
equilibrium  is  reached.  The  proportions  of  liquid 
and  vapor  have  meanwhile  been  altered,  in  this  case 
the  vapor  gaining  at  the  expense  of  the  liquid. 

The  term  "equilibrium"  must  of  course  be  under- 
stood in  a  statistical  sense.  The  balance  is  between  the 
rates  at  which  two  different  molecular  reactions  occur. 
It  is  the  results  of  these  reactions  which  we  consider  as 
opposing.  In  effect  we  use  the  word  " equilibrium" 
much  as  if  the  amounts  of  liquid  and  vapor  formed 
opposite  ends  of  a  lever,  the  position  of  the  fulcrum  rep- 
resenting the  condition  of  equilibrium  between  the  two 

s  257 


258         THE   REALITIES  OF   MODERN   SCIENCE 

phases.  This  position  is  determined  in  the  present  case 
by  the  temperature.  Increasing  the  temperature  means 
shifting  the  equilibrium  in  a  direction  corresponding 
to  increased  vapor. 

In  an  electrolyte  at  a  given  temperature  there  is 
equilibrium  between  the  rate  at  which  molecules  are 
dissociating  and  that  at  which  they  are  recombining 
due  to  collisions.  Let  us  represent  the  molecule  by  AC, 
where  A  stands  for  anion  and  C  for  cation.  There  are 
taking  place  in  the  solution  two  reactions,  AC — >-A  +  C, 
and  A-fC — *-AC  or  AC-* — A-j-C,  that  is,  the  dissocia- 
tion of  molecules  and  the  recombination  of  ions.  Let 
the  rates  be  expressed  in  molecules  per  second  as 
kd  and  fc,  respectively.  When  the  solute  is  added 
to  the  solvent,  kc  is  zero  and  kd  is  a  maximum.  As 
ionization  occurs  the  number  of  normal  molecules 
decreases  and  hence  kd,  the  number  ionized  per  second, 
also  decreases.  On  the  other  hand  kc  increases.  When 
the  two  rates  become  equal  equilibrium l  results. 
Thereafter  dissociation  and  recombination  occur,  but 
the  average  number  of  dissociated  molecules  remains 
constant  unless  we  change  some  of  the  conditions, 
e.g.  the  temperature  or  the  concentration. 

The  rate  of  recombination  will  obviously  depend 
upon  the  frequency  of  collision  of  the  ions  which  are 
necessary  to  form  a  molecule,  and  hence  upon  the 
concentration  of  each  of  these  ions.  Thus  in  the  present 

1  In  the  case  of  a  solution  the  equilibrium  is  reached  very  quickly 
provided  that  the  solute  is  either  in  a  liquid  form  or  finely  powdered. 
If  the  solute  is  in  the  solid  form  it  is  evident  that  but  few  of  its  mole- 
cules are  accessible  for  reaction  with  those  of  the  solvent,  and  hence 
that  the  value  of  kd  will  be  limited  by  the  supply  of  dissolved 
molecules. 


EQUILIBRIA   AND   THEIR   DISPLACEMENT      259 

case  if  we  represent  the  concentrations  of  the  ions  by 
(A)  and  (C)  respectively,  we  have  fccoc  (A)(C).  For 
example,  if  there  are  no  ions  of  one  type  present  there 
can  be  no  recombinations,  and  kc  is  zero.  On  the 
other  hand,  doubling  the  number  of  either  kind  of 
ion  will  double  the  number  of  collisions  of  this  kind 
of  ion  with  the  other. 

In  the  same  way  the  rate  at  which  molecules  dis- 
sociate will  be  proportional  to  the  concentration  of  the 
normal  molecules.  Thus  kd  oc(AC),  where  the  brackets 
represent  the  concentration  of  the  substance  included 
by  them. 

In  equilibrium  kd  and  kc  are  equal.  Let  the  cor- 
responding value  of  the  concentrations  be  represented 
by  a  subscript  e.  Then,  since  kd/kc  =  1  we  have 


(1) 

In  this  particular  case  the  ratio  K,  which  is  called  the 
"equilibrium  constant,"  is  a  constant  expressing  the 
equilibrium  conditions  for  ionization  and  is  therefore 
frequently  called  the  "  ionization  constant." 

The  idea  that  the  rate  at  which  a  substance  reacts 
is  proportional  to  its  molecular  concentration  was 
formulated  in  a  law  by  Guldberg  and  Waage  in  1864. 
We  may  express  it  in  a  general  form  by  saying  that  if 
A  and  B  are  the  reactants  in  a  reaction  which  gives 
the  products  C  and  D,  then  we  may  write  the  rate  at 
which  the  reactants  disappear  as  ki(A)(B)  or  at  the 
equilibrium  point  as  ki(A)e(B)e.  Similarly  if  the 
substances  C  and  D  may  enter  into  a  reverse  reaction 
then  the  rate  at  which  they  disappear  is  fe(C)(D)  or 
at  the  equilibrium  point  k2(C)e(D)e.  At  the  equilib- 


260         THE   REALITIES  OF   MODERN  SCIENCE 

rium  point  the  rate  of  disappearance  of  A  and  B 
must  be  equal  to  the  rate  at  which  they  are  formed  by 
the  disappearance  of  C  and  D.  Therefore 


where  K  is  the  equilibrium  constant  of  the  reaction. 

This  law  is  commonly  known  as  the  law  of  "mass 
action  "  because  originally  the  words  "active  mass" 
were  used  in  place  of  the  modern  term  "concentration." 
The  equilibrium  constant  will  depend  upon  the  tem- 
perature and  may  also  depend  upon  pressure.  For 
this  reason  whenever  an  equilibrium  constant  is  given 
in  numerical  value  it  is  necessary  to  state  the  other 
conditions.  For  very  many  reactions,  on  the  other 
hand,  no  equilibrium  point  may  be  reached,  for  they  are 
not  reversible.  For  example,  there  is  no  equilibrium 
point  for  a  liquid  above  its  critical  temperature,  and 
if  this  temperature  is  exceeded  the  reaction  occurs  in 
one  direction  only  and  goes  to  completion. 

In  the  reactions  of  electrolytes  this  law  of  mass 
action  is  of  'greatest  convenience,  since  the  ionization 
constants  may  be  easily  measured  and  tables  of  them 
prepared.  These  data,  with  tables  of  solubility,  permit 
the  chemist  to  foretell  the  results  of  the  reactions  of 
mixtures  of  electrolytes  and  hence  to  make  efficient 
use  of  them  in  the  quantitative  analysis  of  unknown 
mixtures. 

If  both  components  of  such  a  mixture  are  soluble  and 
also  have  appreciable  dissociations,  chemical  reactions 
are  possible,  that  is,  neutral  molecules  may  be  formed 
by  the  collisions  of  the  anion  of  one  component  with  the 


EQUILIBRIA    AND    THEIR   DISPLACEMENT      261 

cation  of  the  other.  If  these  resultants  are  also  highly 
soluble  and  ionizable  there  will  be  no  evident  chemical 
reaction.  Suppose,  however,  that  one  resultant  is 
practically  insoluble.  In  so  far  as  it  is,  it  will  be  pre- 
cipitated and  thus  remove  two  of  the  products  of  the 
ionization  of  the  original  electrolytes.  This  will  dis- 
place the  equilibrium  and  will  result  in  further  ioni- 
zation. These  new  ions  will  collide  sooner  or  later 
and  form  the  insoluble  product.  This  process  con- 
tinues until  all  the  ions  of  one  kind  derived  from  one 
of  the  original  substances  have  been  removed  from 
the  solution. 

On  the  other  hand,  suppose  that  one  of  the  re- 
sultants is  soluble  but  is  almost  negligibly  ionized. 
So  far  as  removing  ions  from  participation  in  those 
combinations  and  dissociations,  which  would  occur  if 
the  cross  products  were  highly  soluble  and  ionizable, 
this  inability  to  dissociate  is  effective  in  the  same  way 
as  is  insolubility. 

The  extreme  instance  of  the  displacement  of  the 
equilibrium  due  to  the  failure  of  one  of  the  cross  prod- 
ucts to  ionize  is  found  in  the  neutralization  of  an 
acid  by  a  base.  By  definition  an  acid  has  a  cation  of 
H,  and  a  base  has  an  anion  of  OH.  The  two  combine 
to  form  water.  Strictly  speaking,  water  is  not  en- 
tirely free  from  ionization,  but  the  amount  possible 
is  for  all  ordinary  purposes  entirely  negligible.  The 
neutralization1  of  an  acid  by  a  base  therefore  results 

1  Certain  substances  known  as  indicators  are  peculiarly  useful 
in  quantitative  work  for  determining  whether  or  not  neutralization 
has  been  complete.  For  example,  if  it  is  desired  to  determine  the 
percentage  of  a  given  acid  in  an  unknown  mixture,  the  acid  may  be 


262         THE   REALITIES  OF   MODERN  SCIENCE 

in  water  as  one  product  and  a  salt  for  the  other.  (The 
latter,  if  soluble,  is  present  partly  as  dissolved  mole- 
cules and  partly  as  ions,  namely  the  anion  of  the  acid 
and  the  cation  of  the  base.) 

We  have  already  mentioned  the  equilibrium  be- 
tween the  two  phases,  liquid  and  vapor,  of  a  system 
with  only  one  component,  e.g.  water.  We  have  seen 
that  the  condition  is  determined  by  two  variables, 
namely  the  temperature  and  the  volume.  If  either  of 
these  two  conditions  is  altered  the  equilibrium  is  dis- 
turbed and  the  proportions  of  water  and  vapor  are 
varied.  Thus  an  increase  of  temperature  causes  an 
increased  vaporization  and  a  consequent  reduction,  of 
the  portion  which  the  water  is  of  the  entire  volume. 
Similarly  if  the  volume  is  varied,  as  for  example  de- 
creased, there  is  a  change  in  the  proportions  of  the  two 
phases,  vapor  being  condensed. 

In  both  cases  we  notice  that  the  change  is  of  such  a 
nature  as  to  oppose  the  cause.  Thus  in  the  case  of 
increased  temperature  there  is  an  increase  in  the 
kinetic  energy  of  the  molecules  of  the  system,  and  this 
results  in  a  separation  of  more  molecules  from  the 
liquid.  This  increased  molecular  separation  means  a 
greater  potential  energy  for  the  molecules.  In  so  far 
as  the  potential  energy  is  increased  the  kinetic  energy  is 
reduced.  The  change  occurring  is  therefore  of  such 
a  character  as  to  reduce  the  molecular  kinetic  energy 

neutralized  by  the  addition  of  a  base  of  known  concentration  and 
hence  known  ionization.  An  indicator,  e.g.  methyl  orange,  litmus, 
or  phenolphthalein,  is  used  to  show  when  the  amount  of  the  base 
which  is  being  added  is  just  sufficient  to  neutralize  all  the  H  ions. 
The  amount  of  acid  may  then  be  calculated  from  the  required 
amount  of  the  base. 


EQUILIBRIA   AND    THEIR   DISPLACEMENT      263 

and  hence  the  temperature.  Similarly,  in  the  case  of 
the  reduced  volume,  there  is  a  decrease  in  the  molecu- 
lar potential  energy,  which  is  thereby  converted  into 
kinetic  energy  and  thus  increases  the  pressure  exerted 
by  the  vapor  molecules.  This  change  or  reaction  is 
also  such  as  to  oppose  the  action. 

In  Newton's  third  law  of  motion  we  have  the  expres- 
sion of  the  physical  fact  that  every  action  is  accom- 
panied by  an  equal  and  opposite  reaction.  We  are 
accustomed,  however,  to  consider  this  generalization 
as  referring  specifically  to  the  interactions  of  matter  in 
bulk.  In  an  earlier  chapter  we  met  the  same  general- 
ization, as  expressed  by  Lenz,  for  the  interactions  of 
moving  electrons.  In  the  case  of  molecular  reactions 
this  important  idea  awaited  expression  until  the  work 
of  Le  Chatelier  in  1884  and  Braun  in  1887. 

Their  principle  is,  that  any  external  action  upon  a 
system  produces  a  change  of  such  a  character  (in  such 
a  sense)  that  the  resistance  which  the  system  offers 
to  the  external  action  is  increased.  In  quantitative 
form  this  principle  is  embodied  in  the  second  law  of 
thermodynamics.  The  latter,  however,  involves  some 
of  the  most  difficult  abstract  reasoning  to  be  found 
in  physical  science.  Its  mathematical  expression  also 
does  not  lend  itself  to  simple  verbal  interpretation  hi 
elementary  treatments  of  science.  In  the  principle  of 
Le  Chatelier-Braun,  however,  we  are  fortunate  in 
having  a  non-mathematical  expression  of  marked 
simplicity  which  indicates  the  character  of  the  reac- 
tions, the  quantitative  relations  of  which  are  expressed 
by  the  mathematical  forms  of  the  second  law.  This 
principle  is  not  restricted  by  any  assumptions  as  to 


264         THE   REALITIES   OF   MODERN  SCIENCE 

molecular  structure,  and  like  the  second  law  is  per- 
fectly general.  It  serves  admirably  either  in  lieu  of 
this  law  for  students  whose  mathematical  training  is 
insufficient,  or  as  an  introduction  to  the  law. 

From  our  study  of  continuity  of  state  we  recognize 
that  the  molecular  state  of  a  body  or  system  is  deter- 
mined by  various  conditions,  of  which  temperature, 
pressure,  and  volume  are  the  three  so  far  considered.1 
These  conditions,  or  parameters,  are  variable  at  the 
will  of  the  experimenter.  The  behavior  of  the  mole- 
cules of  the  substance  is  not  directly  controllable, 
however,  but  only  indirectly  as  one  or  the  other  of 
these  factors  may  be  varied  by  an  external  action. 
The  principle  of  Le  Chatelier-Braun  enables  us  to  pre- 
dict in  any  case  what  the  character  of  the  effect  upon 
the  molecules  will  be  when  one  of  these  parameters 
is  varied. 

For  example,  consider  the  case  of  a  gas  under  a  given 
pressure  and  occupying  a  definite  volume  at  a  definite 
temperature.  If  the  external  pressure  is  increased,  the 
effect  is  to  decrease  the  volume.  The  equilibrium 
which  the  moment  before  existed  between  the  external 
pressure  and  the  opposing  pressure  of  the  gas  molecules 
has  been  disturbed.  The  volume  is  reduced.  Now, 
the  principle  tells  us  the  compensating  change  which 
occurs  within  the  system.  The  change  in  V  is  ac- 
companied by  a  change  in  one  of  the  other  variables, 
in  this  case  T,  of  such  a  character  as  to  oppose  the 
change  in  V.  That  gases  expand  with  increased  tem- 

1  Electromotive  force,  that  is,  a  difference  in  electrical  potential 
energy,  is  obviously  another  factor.  The  ions  which  take  part  in 
chemical  reactions  do  so  as  the  result  of  the  potential  energy  of  the 
electronic  systems  which  they  form  with  one  another. 


EQUILIBRIA   AND    THEIR   DISPLACEMENT      265 

perature,  is  an  experimental  fact,  independent  of  any 
theories  as  to  molecular  behavior.  If  the  change  hi 
temperature  is  to  oppose  a  decrease  in  volume  the 
temperature  must  increase.  That  the  temperature  of 
a  gas  increases  as  the  volume  is  decreased  is  the  simple 
observation  of  every  one  who  has  pumped  a  bicycle 
or  automobile  tire. 

We  may  express  this  principle,  then,  by  saying  that 
when  a  parameter,  e.g.  x,  which  is  one  of  those  deter- 
mining the  equilibrium  of  a  system,  is  altered,  another 
parameter,  e.g.  y,  changes  in  such  a  manner  as  to 
diminish  the  direct  effect  of  the  external  cause.  It  is, 
of  course,  necessary,  if  we  are  to  foretell  the  result 
of  a  change  in  x,  that  we  should  know  how  x  and  y 
are  related.  In  the  particular  case  just  studied  we 
know  that  V  is  proportional  to  T.  The  change  in 
T  must  be  that  corresponding  to  a  change  in  V  opposite 
to  that  which  actually  takes  place.  Hence,  since  V 
actually  decreases,  T  must  increase. 

The  principle  is  not,  however,  limited  in  its  applica- 
tions to  the  phenomena  of  gases  and  vapors.  Con- 
sider the  case  of  an  elastic  wire  at  a  definite  tempera- 
ture and  having  a  certain  length  under  a  definite 
tension.  What  will  be  the  effect  of  stretching  the  wire 
still  further?  The  parameter  which  is  changed  by 
the  external  cause  is  the  length.  The  other  parameter 
is  the  temperature.  Now,  we  know  that  metal  wires 
are  elongated  by  an  increase  in  temperature.  The 
increase  in  length  will  then  be  accompanied  by  a 
change  in  temperature  in  such  a  direction  as  would 
normally  produce  the  opposite  effect.  Hence  the  wire 
is  cooled  by  stretching. 


266         THE   REALITIES  OF   MODERN  SCIENCE 

Consider  for  example  an  illustration  in  the  case  of 
surface  tension  which,  we  have  seen,  decreases  with  an 
increase  in  temperature.  If  the  surface  of  a  liquid  is 
increased,  what  will  be  the  nature  of  the  effect  on  the 
temperature  of  the  liquid?  The  temperature  must 
change  in  such  a  direction  as  to  increase  the  surface 
tension  and  thus  to  resist  the  increase  in  surface  area. 
Hence,  the  temperature  must  decrease.  From  our  con- 
siderations of  molecular  potential  energy  (see  page  221) 
we  arrived  at  the  same  conclusion.  Thus,  an  in- 
crease in  surface  means  an  increase  of  p.e.  due  to  the 
increased  separation  of  the  molecules,  and  hence  a  de- 
crease in  the  k.e.,  that  is,  a  decrease  in  the  temperature. 

Suppose  the  temperature  of  a  two-phase  system, 
e.g.  water  and  ice,  is  kept  constant,  how  will  the  equi- 
librium be  affected  by  the  introduction  of  a  soluble 
solid,  e.g.  a  salt?  The  salt  dissolves,  in  part  at  least. 
Its  introduction  increases  the  concentration  of  salt 
molecules  from  the  original  value  of  zero.  The  change 
which  takes  place  in  the  mixture  must  then  be  of  such  a 
character  as  to  oppose  this  change  in  concentration. 
The  equilibrium  between  the  ice  and  the  water  is, 
therefore,  altered  in  such  a  manner  as  to  tend  to  reduce 
the  concentration  of  the  salt  solution.  Hence  the  ice 
melts  so  as  to  increase  the  amount  of  the  solvent  water. 

An  interesting  comparison  has  been  suggested  be- 
tween the  principle  of  Le  Chatelier-Braun  and  the 
principle  in  organic  life  which  is  illustrated  by  the 
" faculty  of  accommodation."  This  is  the  faculty  by 
virtue  of  which  external  actions  upon  a  living  organism 
produce  changes  which  tend  to  increase  its  power  of 
resistance  to  such  external  actions.  Of  this  phe- 


EQUILIBRIA    AND    THEIR   DISPLACEMENT      267 

nomenon  the  development  of  muscle  by  exercise  is  a 
familiar  illustration.  We  might,  therefore,  look  upon 
the  principle  of  Le  Chatelier  as  an  extension  to  in- 
animate substances  of  the  faculty  of  accommodation. 

We  may  also  express  the  principle  by  saying  that 
every  system,  chemical  or  physical,  which  is  in  equi- 
librium is  conservative,  that  is,  tends  to  remain  un- 
changed. In  the  world  about  us,  however,  changes 
are  constantly  occurring  in  the  molecular  systems. 
We  conclude,  therefore,  that  they  are  not  in  equilib- 
rium. Certain  transformations  are  taking  place  in 
these  systems  naturally,  that  is,  without  external  cause. 
Such  a  system,  for  example,  is  the  solar  system,  in  which 
we  live,  where  the  transformations  take  place  "of 
themselves, "  as  we  might  say. 

What  are  the  typical  transformations  which  take 
place  naturally?  We  have  progressed  far  enough  hi 
our  study  to  realize  that  the  production  of  molecular 
motions,  or  "heat,"  at  the  expense  of  work  is  a  natural 
process,  as  in  the  phenomenon  of  friction.  Similarly 
the  mixing  of  gases  by  diffusion  is  a  natural  process. 
The  transfer  of  energy  from  molecules  with  high  kinetic 
energy  to  those  of  low  is  also  natural,  that  is,  the  transfer 
of  heat  from  a  hot  body  to  a  cold  body.  The  radia- 
tion of  energy  of  which  light  is  an  example  is  another 
natural  process. 

What  is  the  result  of  these  natural  transformations? 
We  notice  that  they  tend  toward  a  dead  level  of  mo- 
lecular energy.  The  faster  moving  molecules  lose 
energy  to  the  slower.  The  final  condition  of  equilib- 
rium toward  which  the  universe  tends  hi  these  natural 
processes  is  then  one  of  uniform  temperature.  The 


268         THE   REALITIES  OF   MODERN   SCIENCE 

temperature  will  be  the  lowest  possible.  The  ratio 
of  all  the  energy  in  the  universe  to  its  absolute  tem- 
perature will  then  be  as  large  as  it  can  become.  Now 
this  quotient  of  heat  energy  and  absolute  temperature 
is  usually  called  " entropy."  The  natural  processes 
or  transformations  of  the  universe  tend  to  increase 
the  entropy. 

The  processes  mentioned  above  are  natural  much  as 
it  is  natural  for  water  to  run  down  hill.  We  can  make 
water  go  up  hill,  but  it  is  not  natural  and  we  have  to  do 
work.  In  the  same  way  we  can  separate  the  molecules 
of  gases  which  have  mixed  by  diffusion,  we  can  produce 
work  from  heat,  and  we  can  transfer  energy  from  cold 
bodies  to  hot  ones,  making  the  former  colder  and  the 
latter  hotter,  but  these  are  not  natural  processes.  In 
every  instance  we  have  to  do  work  to  accomplish  the 
result.  We  can  only  do  the  necessary  work  at  the 
expense  of  some  other  energy  transformation  which 
is  natural.  Those  processes,  which  are  the  reverse 
of  the  natural  ones,  have  never  been  known  to  occur  of 
their  own  accord  and  scientists  are  agreed  in  believing 
that  they  never  will.  They  may  be  produced,  but 
only  when  a  transformation  which  is  natural  supplies 
the  energy. 

These  statements  are  the  sense  of  the  Second  Law  of 
Thermodynamics.  The  mathematical  statement  of  the 
law  is  usually  made  in  terms  of  that  ratio  of  molecular 
kinetic  energy  or  heat  to  absolute  temperature  which 
we  have  just  called  "  entropy."  The  law  says  that 
all  natural  transformations  in  the  universe  result  in  an 
increase  in  entropy.  The  student  of  thermodynamics, 
however,  finds  it  a  great  convenience  to  reason  about 


EQUILIBRIA    AND    THEIR   DISPLACEMENT      269 

ideal  and  limiting  cases  of  transformations  where  there 
is  no  increase  in  entropy.  The  most  famous  case  is 
known  as  the  Carnot  cycle  which  is  typical  of  heat 
engines  or  internal  combustion  engines.  The  conclu- 
sion reached  by  Carnot  and  by  Clausius  is  that  the 
maximum  efficiency  of  such  a  cycle  is  independent  of 
the  working  substance,  e.g.  water,  or  air,  and  depends 
only  upon  the  extremes  of  temperature  to  which  the 
substance  is  subjected.  The  efficiency  will  be  much 
less  than  this  in  all  but  the  ideal  case. 


CHAPTER  XX 

MOLECULAR  MAGNITUDES 

EVIDENCE  as  to  the  reality  of  the  molecular  and 
atomic  structure,  which  we  have  been  discussing,  is 
to  be  found  in  the  agreement  between  the  values 
obtained  for  various  molecular  magnitudes  from  en- 
tirely different  experimental  determinations.  The 
most  important  magnitudes  are  (1)  the  number  per 
mole  under  standard  conditions  of  pressure  and  temper- 
ature, (2)  the  mass  of  each  molecule,  (3)  its  diameter, 
and  (4)  its  kinetic  energy  of  translation.  A  knowl- 
edge of  these  is  desirable  in  order  to  obtain  a  fairly 
complete  picture.  In  addition  we  shall  need  to  consider 
later  the  total  energy  possessed  by  a  molecule  at  a 
given  temperature  and  its  partition  among  the  degrees 
of  freedom. 

Many  attempts  have  been  made  to  determine  the 
number  of  molecules  in  one  mole.  The  earlier  values 
were  obtained  indirectly  from  an  estimate  of  the 
diameter  of  a  molecule.  For  example,  Lord  Rayleigh 
made  use  of  the  fact  that  the  surface  tension  of  a  liquid 
is  greatly  affected  by  the  presence  of  impurities.  A 
particle  of  camphor  placed  on  a  water  surface  jerks 
about  due  to  variations  in  the  surface  tension  about 
it.  Oil  spread  on  the  water  reduces  these  motions  and 
Rayleigh  determined  the  critical  thickness  of  an  oil 

270 


MOLECULAR   MAGNITUDES  271 

film  which  would  just  cause  them  to  cease.  He  found 
that  a  layer  of  approximately  1.6xlO~7  cm.  was 
sufficient.  Since  the  layer  cannot  be  less  than  one 
molecule  in  thickness  the  diameter  of  a  molecule  is 
not  greater  than  this  distance. 

Somewhat  similar  values  were  obtained  by  various 
investigators  using  soap  films.  We  have  all  noticed 
that  as  a  soap  bubble  drains,  its  iridescent  colors  change 
and  a  black  spot  appears  at  the  point  where  it  ultimately 
breaks.  Measurements  of  this  thickness  are  possible 
by  optical  means  and  another  value  of  the  upper 
limit  for  the  diameter  of  a  molecule  has  thus  been 
obtained. 

Some  of  the  most  consistent  values  of  the  number  of 
molecules  per  mole  were  obtained  by  Pen-in.  He  used 
small  granules  of  gamboge  or  of  mastic,  substances 
which  do  not  dissolve  in  water  but  form  emulsions. 
Such  particles  he  sorted  into  uniform  size  by  centri- 
fuging.  The  average  kinetic  energy  of  these  is  the 
same l  as  that  of  a  molecule  of  the  liquid  in  which  they 
are  placed.  But  it  is  this  energy  which  accounts  for 
the  osmotic  pressure  and  brings  about  a  diffusion  of 
the  particles  when  the  liquid  is  not  agitated. 

In  a  liquid  the  osmotic  pressure  will  depend  at 
each  point  upon  the  concentration  of  the  particles.  If 
there  is  no  external  cause  like  gravity  the  concentration 
will  be  the  same  at  all  points  when  the  liquid  has 
reached  a  state  of  equilibrium.  Because  of  gravity, 

1  That  is,  there  is  an  equipartition  of  k.e.  between  the  particles 
and  the  molecules  of  the  liquid.  It  is  this  equipartition  which 
makes  such  Brownian  movements  so  valuable  in  the  study  of  the 
invisible  molecules. 


272         THE   REALITIES  OF   MODERN   SCIENCE 

however,  the  particles  in  Perrin's  experiment  fell  until 
the  concentration  at  each  level  in  the  liquid  became 
such    that    a    further    decrease    in    the    gravitational 
p.e.  of  the  particles  would  have  occa- 
.*."'•-       '•      sioned  a  greater  increase  in   osmotic 
*.'.,."    "•'•*'    energy  due  to  the  increased  concentra- 
...    '.    **/         tion.     Figure  32  represents  a  vertical 
„.•;'•'      '•'.        section  of  an  emulsion  in  equilibrium. 
Perrin,  therefore,  was  able  to  obtain  a 
relation  between  the  decrease  in  gravi- 
tational energy,  occasioned  by  a  par- 
ticle in  moving  from  one  level  to  a  new 
level,  h  cm.  lower,  and  the  correspond- 
ing  change  in  the  energy  which  the 
particle  possessed  as  a  result  of  its  in- 
clusion in  a  group  of  greater  concen- 
tration and  hence  of  greater  osmotic 
pressure. 

If  each  particle  is  spherical,  of  radius 
r  and  of  density  D,  its  mass  is  4?rZ)r3/3. 
The  decrease  in  gravitational  potential 
energy  of  the  particle  considered  by 
itself  is  4(7rr3D%)/3.  Its  movement 
downward  results  in  the  movement 
upward  of  a  similar  volume  of  the 
liquid,  that  is,  it  may  be  considered  to 
change  places  with  an  equal  volume  of 
the  liquid  of  density  d,  which  therefore  gains  in  poten- 
tial energy  the  amount  4(7rr3d%)/3. 

The  corresponding  increase  in  energy  on  the  part  of  the 
average  particle  is  expressed  as  KwT,  where  wT  is  the 
average  kinetic  energy  of  translation  of  a  liquid  mole- 


MOLECULAR   MAGNITUDES  273 

cule.  The  factor  K  involves  the  ratio1  of  the  concen- 
trations at  the  two  levels,  and  hence  may  be  expressed 
in  terms  of  the  number  of  particles  in  equal  areas  at 
these  levels.  These  numbers,  n  and  n0,  Perrin  deter- 
mined by  direct  count,  viewing  different  layers  of  the 
emulsion  through  a  microscope. 

The  kinetic  energy  of  translation,  wT,  may  be  ex- 
pressed by  using  the  perfect  gas  equation,  which  we 
saw  from  our  study  of  osmosis  is  applicable  to  such 
cases.  From  page  218  we  have  w  =  3R/2N.  Using 
these  values  of  K  and  w  in  the  expression  of  the  energy 
relations  at  equilibrium,  gives 

(1) 

where  N  is  the  number  of  molecules  per  mole.  The 
other  terms  on  the  right-hand  side  of  this  equation  are 
known.  The  density,  d,  was  easily  measured.  The 
density,  Z>,  of  the  mastic  Perrin  determined  from  a 
solid  piece  before  forming  the  emulsion  and  checked 
after  the  main  experiment  by  evaporating  the  liquid 
and  measuring  the  density  of  the  residue.  The  dis- 
tance, h,  was  directly  measured.  There  remained  to 
be  determined  only  the  radius  r.  The  granules  which 
he  used  were  about  2X10~5  cm.  in  diameter,  much  too 
small  for  an  accurate  direct  measurement.  The  radius 
was  obtained  by  making  use  of  an  equation  developed 
by  Stokes  which  states  the  rate  at  which  small  spherical 

1  This  factor,  which  is  conveniently  derived  by  using  integral 
calculus,  involves  the  logarithm  of  the  ratio  no/n.  The  expression 
is  K  =  (2/3)  loge(noAO-  A  "calculus  dodging"  derivation  is  given 
by  Perrin  in  "  Les  Atomes,"  an  interesting  book,  published  in  trans- 
lation by  D.  Van  Nostrand  Company,  1916. 
T 


274         THE   REALITIES  OF   MODERN   SCIENCE 

particles  will  fall  under  gravity  through  a  viscous 
fluid.  The  emulsion  was  therefore  shaken  until  thor- 
oughly mixed  and  a  measurement  was  made  of  the  rate 
at  which  the  upper  layers  cleared  of  particles. 

The  phenomenon  of  viscosity,  sometimes  spoken  of 
as  molecular  friction,  is  most  easily  described  in  the 
case  of  the  flow  of  a  liquid  through  a  pipe.  Suppose 
the  liquid  wets  the  walls,  then  the  layer  of  molecules 
immediately  adjacent  to  the  walls  is  at  rest  and  the 
next  layer  slides  by  it.  In  terms  of  the  kinetics  of 
molecules  we  say  that  there  is  now  a  "mass  motion" 
in  addition  to  the  haphazard  molecular  motion.  The 
mass  motion  of  the  inner  streaming  layer  is,  however, 
interfered  with  by  the  absence  of  such  motion  on  the 
part  of  the  outer  layer.  The  molecular  motion  of  the 
streaming  molecules  causes  some  of  them  to  collide 
with  those  of  the  fixed  layer  and  they  are  thus  retarded. 
Layers  farther  from  the  stationary  layer  will  be  re- 
tarded less  by  such  collisions.  (The  effect,  being  due 
to  collisions  and  not  to  attractions,  takes  place  in  all 
aeriform  substances,  including  perfect  gases.  It  is 
viscosity  which  retards  the  motion  of  bodies  through 
the  air  and  of  which  we  have  spoken  as  air  friction.) 
In  terms  of  force  it  is  measured  by  the  number  of  dynes 
which  must  be  applied  tangentially  to  each  square 
centimeter  of  the  surface  of  one  layer  of  the  fluid  to 
cause  this  layer  to  move  with  a  velocity  of  1  cm.  per 
second  with  reference  to  a  parallel  layer  which  is  1  cm. 
away.  This  ratio  is  the  coefficient  of  viscosity.  This 
quantitative  definition  is  concerned  only  with  relative 
velocities  and  is  not  limited  to  the  special  case  which 
we  described  where  one  layer  is  at  rest. 


MOLECULAR   MAGNITUDES 


275 


For  the  case  of  a  sphere  it  was  calculated  by  Stokes 
that  the  reaction  due  to  viscosity  is  GTJTCV,  where  r 
is  radius,  v  is  the  velocity,  and  c  is  the  coefficient  of 
viscosity  of  the  fluid.  The  acting  force  which  occasions 
the  fall  is  the  space  rate  of  change  of  the  energy  and 
therefore  equals  the  quotient  of  the  left-hand  side  of 
equation  (1)  and  h.  Hence  equating  action  and 
reaction  gives 

(4/3)irr»(D  -  d)g  =  farm  (2) 


The  coefficient  c  is  obtainable  from  tables  of  previous 
determinations  or  may  be  found  by  a  separate  experi- 
ment. Stokes'  Law,  therefore,  enabled  Perrin  to 


FIG.  33. 


determine  r  from  his  observations  of  v.  Substitution 
of  this  value  in  equation  (1)  then  gave  N.  By  this 
experiment  he  found  Ar  =  6.83Xl023. 

Although  this  method  makes  use  of  the  kinetics  of 
visible  particles  it  considers  the  average  particle  rather 
than  the  Brownian  movement  of  a  particular  particle. 


276         THE   REALITIES  OF   MODERN  SCIENCE 

Another  series  of  Perrin's  experiments  dealt  with  in- 
dividual motions.  If  one  of  the  particles  is  observed 
in  a  microscope  its  motion  will  be  of  the  character 
illustrated  by  the  diagrams  of  Fig.  33.  Its  position 
is  observed  and  recorded  every  t  seconds  (t  was  30  in 
this  instance).  It  is  by  such  irregular  motions  that 
the  diffusion  of  the  particles  takes  place.  Einstein 
had  shown  a  relation  between  the  coefficient  of  diffusion 
and  the  average  square  of  the  displacements  of  a 
particle.  The  diffusion  is  expressible  in  terms  of 
RT/N  and  the  viscosity  of  the  liquid  if  the  radius  of 
the  particle  is  known.  If  X2  represents  the  average 
square  of  the  displacements  along  the  x-axis  for  times 
of  value  t  seconds,  then  it  was  shown  that 

(3) 


where  the  other  terms  have  their  previous  meanings. 
Perrin  observed  several  thousand  displacements  and 
obtained  as  his  average  value  for  N,  6.88  X1023. 

The  most  reliable  determination  of  the  number  of 
molecules  in  a  mole,  is  the  work  of  Millikan.  It  is 
essentially  a  by-product  of  his  determination1  of  the 
amount  of  electricity  represented  by  an  electron. 
You  will  remember  that  the  electrostatic  unit  of  quan- 
tity was  defined  (on  page  174)  in  terms  of  the  repulsion 
of  two  unit  charges  at  unit  distance,  and  the  electro- 
magnetic unit  (on  page  206)  in  terms  of  the  repulsion 
exerted  on  a  unit  magnetic  pole  at  the  center  of  a  unit 
arc  of  unit  current.  The  practical  unit  of  current  is 
the  ampere  and  is  nominally  one  tenth  of  the  latter 
unit.  Because  of  its  importance  in  the  arts  the  ampere 
1  To  be  described  in  Chapter  XXII. 


MOLECULAR    MAGNITUDES  277 

has  been  legally  defined  as  the  steady  current  which  in 
1  second  will  transfer  1  coulomb  of  electricity.  The 
coulomb,  which  is  defined  as  one  tenth  of  the  quantity 
transferred  in  1  second  by  one  absolute  e.m.  unit  of 
current,  is  legally  defined  as  the  quantity  of  electricity 
which  is  transferred  in  electrolysis  between  two  silver 
plates  in  a  slightly  acid  solution  of  AgNOs  when 
0.001118  gram  of  silver  is  deposited  on  the  cathode. 
Since  the  atomic  weight  of  Ag  is  commonly  accepted 
to  be  107.88,  it  follows  that  the  deposition  of  1  mole 
of  silver  represents  the  transfer  of  107.88/0.001118  or 
96500  coulombs,  or  9650  absolute  e.m.  units. 

The  value  of  the  charge  carried  by  the  electron  is, 
however,  usually  expressed  in  electrostatic  units. 
Represent  the  value  of  the  electron  in  these  e.s.  units 
as  e.  The  e.m.  unit  of  quantity  is  3X1010  times  the 
e.s.  unit,  as  may  be  determined  experimentally  by 
measuring  the  same  charge  of  electricity  first  in  one  unit 
and  then  in  the  other.1  By  1  mole  of  silver  there  is, 
then,  transferred  9650(3  X1010)  or  28,950 X1010  abso- 
lute electrostatic  units  of  electricity.  If  N  represents 
the  number  of  silver  atoms  (strictly  ions)  which  are  in 
the  mole,  then  in  e.s.  units  this  amount  of  electricity 
is  Ne.  Millikan's  value  for  e  is  4.774  X10~10,  and  hence 
his  value  for  N  is  60.65  X1022. 

The  mass  of  a  molecule  of  any  substance  is  to  be 
found  as  the  quotient  of  its  molecular  weight  in  grams 
and  N.  Thus  for  hydrogen  the  mass  is  2.016/60.65  X 1022 

1  The  ratio  is  the  velocity  of  light.  This  experimental  fact  was 
one  piece  of  evidence  substantiating  Maxwell's  theories,  which  were 
referred  to  on  p.  124  as  anticipating  Hertz's  discovery  of  electro- 
magnetic waves  of  frequencies  much  smaller  than  those  of  light. 


278         THE   REALITIES   OF   MODERN   SCIENCE 

or  3.33X10"24  gram  per  molecule.  For  oxygen  it  is 
32/2.016  times  as  large  or  52.8  X  10~24  gram  per  molecule. 
The  root  mean  square  velocity  of  a  gaseous  molecule 
may  be  obtained  without  a  knowledge  of  N  by  con- 
sidering equation  (2)  of  page  164.  Thus 

p=Nmv2/3V  =  dv*/3  (4) 

where  d  is  the  density.  Then  v  is  the  square  root  of 
3p/d  or 

va  =  0.921*;  =  0.921  VZp/d  (5) 

where  va  is  the  average  velocity.  The  factor,  0.921,  is 
to  be  obtained  only  by  a  mathematical  analysis  be- 
yond the  scope  of  this  book.  For  hydrogen  under 
standard  condition  p  =  1.013Xl06  dynes/sq.cm.,  and 
d  =  2.016/22410  g./c.  c.,  hence  va  =  169,200  cm./sec.,  that 
is,  of  the  order  of  one  mile  a  second. 

The  velocity  of  any  other  gas  will  be  less  than  this, 
being  inversely  as  the  square  root  of  its  molecular 
weight.  Oxygen  will  have  an  average  molecular  veloc- 
ity, under  the  same  conditions  of  temperature  and 
pressure,  of  one  fourth  this  amount  or  42,500  cm./sec. 
If  the  temperature  is  not  0°  C.  the  velocity  will  be  to 
the  velocity  under  standard  conditions  in  the  ratio  of 
the  square  roots  of  the  absolute  temperatures.  Since 
pressure  is  directly  as  density,  the  velocity  will  not 
vary  with  the  pressure. 

The  mean  free  path  of  a  gaseous  molecule  may  be 
obtained  by  measurements  of  its  viscosity.  We  have 
seen  that  a  force  is  required  to  maintain  a  difference 
in  mass  motion  of  two  parallel  layers  of  a  gas  because 
of  the  collisions  which  occur  between  molecules  of 
originally  different  layers  and  hence  of  different  mass 


MOLECULAR   MAGNITUDES  279 

velocities.  The  greater  the  number  of  such  collisions 
the  greater  will  be  the  tendency  of  the  molecules  of 
all  the  layers  to  come  to  a  common  speed  and  hence 
the  greater  must  be  the  force l  which  maintains  the  de- 
sired difference.  Other  things  being  equal  we  should 
expect  that  the  greater  the  density,  the  greater  will 
be  the  viscosity.  If  th?  molecules  of  a  fast-moving 
layer  may  travel,  before  collision,  only  to  a  near-by 
layer,  the  molecules  of  which  are  moving  with  a  speed 
but  slightly  different,  then  the  effect  of  the  collisions 
is  not  so  great  as  it  would  be  if  the  molecules  move 
past  several  layers  and  collide  with  those  of  a  much 
slower  mass  motion.  Similarly  we  expect  the  effect  of 
the  slow  moving  molecules  to  be  less  if  they  do  not 
move  into  far  distant  and  rapidly  moving  layers  and 
hence  that  the  greater  the  mean  free  path  of  the  mole- 
cules the  greater  will  be  the  viscosity.  These  hypothet- 
ical effects  are  not  independent,  for  greater  density 
means  smaller  mean  free  path.  The  effect  will  also 
depend  upon  the  average  molecular  velocity.  It  may 
be  shown  by  reasoning  similar  to  that  followed  hi 
Chapter  XIII,  that 

c  =  vLd/3  (6) 

where  d  is  density,  v  is  average  velocity,  L  is  the  mean 
free  path,  and  c  is  the  coefficient  of  viscosity.  For 
example  at  0°C.  for  hydrogen  c  =  0.0000889,  v  =  169,200, 
d  =  0.0000898,  and  hence  L  =  1.76X10~5  cm. 

1  It  is  evident  that  energy  is  expended  in  maintaining  this  dif- 
ference in  mass  velocity  between  two  layers.  The  external  energy 
imparted  to  the  molecules  in  a  mass  motion  is  thus  seen  to  be  con- 
stantly degrading  into  molecular  kinetic  energy  of  haphazard  mo- 
tion. The  average  molecular  velocity,  and  hence  the  temperature, 
of  the  gas  therefore  increase. 


280 


THE   REALITIES  OF   MODERN   SCIENCE 


Expressions  have  been  derived  for  the  diameter  of 
a  molecule  in  terms  of  its  mean  free  path.  We  shall 
only  indicate  the  line  of  reasoning  which  has  been  fol- 
lowed in  such  derivations.  If  all  the  molecules  in  a 
c.c.  of  gas  are  imagined  to  be  evenly  spaced  there  will 
be  in  each  row  or  column  of  the  cube  ^N0  molecules, 
where  NQ  is  the  total  number  per  c.  c.  The  distance 


FIG.  34. 


between  the  centers  of  two  adjacent  molecules  will 
then  be  D 


Now  imagine  all  the  molecules 
except  one  to  be  stationary  in  this  configuration. 
This  molecule  will  trace  out  a  cylinder  having  the  same 
diameter  as  itself,  as  illustrated  in  Fig.  34.  The 
average  length  of  all  the  possible  circumscribing  cyl- 
inders which  just  fall  short  of  inclosing  a  portion  of 
another  molecule,  is  therefore  the  mean  free  path 


MOLECULAR  MAGNITUDES  281 

of  a  molecule  under  the  condition  that  its  neighbors 
are  stationary. 

Now  Clausius  showed  that 

L'/I>  =  £Y47rr2  (7) 

where  Z/  is  this  mean  free  path,  r  is  the  radius  of  the 
molecule,  and  l/D  is  the  cube  root  of  the  number  of 
molecules  per  c.  c.  As  a  matter  of  fact  all  the  molecules 
are  in  motion  with  different  velocities,  so  that  the 
problem  is  not  quite  as  simple  as  the  case  we  have  just 
considered.  The  mean  free  path  is  shorter  than  L'  and 
is  L'/\/2~.  Hence  if  we  express  the  diameter  in  terms 
of  the  mean  free  path  we  have 


(2r)2  =  D*/7rV2L  =  l/W^VoL  (8) 

Substituting  for  N0  its  value  of  60.65  X1022/22410  and 
for  L  the  value  found  above,  we  find  for  hydrogen 
2r  =  2.17XlO-8cm.  Similarly  for  oxygen  2r  =  2.99XlO~8 
cm. 

There  are  other  methods  by  which  these  molecular 
magnitudes  may  be  obtained.  For  example,  the  con- 
stants of  Van  der  Waals's  equation  furnish  an  indica- 
tion of  the  molecular  diameter.  This  method  is  not, 
however,  as  accurate  as  some  others,  e.g.  that  of  vis- 
cosity discussed  above,  but  it  leads  to  results  of  the 
same  order  of  magnitude.  For  example,  Van  der  Waals 
found  2r  for  hydrogen  to  be  1.04X10"8  cm.,  while  a 
more  recent  determination  using  the  same  method 
more  carefully  gave  1.26X10"8  cm. 

Knowing  the  mean  free  path  and  the  average  velocity 
we  may  calculate  the  average  number  of  collisions  per 
second  as  v/L.  Thus  for  hydrogen  there  are  about 


282         THE  REALITIES  OF   MODERN  SCIENCE 

1010  and  for  oxygen  approximately  half  as  many  col- 
lisions per  second  under  standard  conditions. 

The  mean  translational  kinetic  energy  of  a  mole- 
cule per  degree  is  found  by  the  equation  w  =  3R/2N  as 
3(83.2)  X106/2(60.65)X1022  or  2.06XH)-16  ergs. 

As  an  illustration  of  the  variations  in  the  determi- 
nation of  molecular  magnitudes  and  also  of  the  re- 
markable consistency  of  the  values  obtained  by  widely 
different  methods,  consider  the  following  table l : 

TABLE    II 

Phenomena  Observed N+1Q22 

Viscosity  of  gases  (Van  der  Waals's  equation) 62. 

Brownian  Movement  —  Distribution  of  Grains    ....  68.3 

Brownian  Movement  —  Displacements 68.8 

Brownian  Movement  —  Rotations 65. 

Brownian  Movement  —  Diffusion 69. 

Irregular  Molecular  Distribution  —  Critical  Opalescence   .  75. 

Irregular  Molecular  Distribution  —  The  Blue  of  the  Sky  .  55. 

Black  Body  Spectrum 64. 

Charged  Spheres  (in  a  gas) 68 

Radioactivity  —  Charges  Produced 62.5 

Radioactivity  —  Helium  engendered 64. 

Radioactivity  —  Radium  Lost        71. 

Radioactivity  —  Energy  Radiated 60. 

Millikan's  Value  —  Charged  spheres  in  a  gas 60.65 

The  agreements  are  more  remarkable  than  the  dis- 
crepancies when  we  stop  to  realize  what  an  enormous 
number  of  molecules  there  are  in  a  mole.  The  funda- 
mental assumptions  of  the  kinetic  theory  cannot  be 
doubted  in  the  light  of  this  evidence.  This  does  not 
mean,  however,  that  certain  assumptions  which  are 
sometimes  made  for  convenience  of  mathematical 
analysis  are  verified.  For  such  analysis  it  is  usual  to 
assume  that  the  molecules  behave  like  hard  elastic 

1  From  Perrin's  "Atoms"  except  for  the  addition  of  Millikan's 
value  and  the  value  obtained  from  the  blue  of  the  sky. 


MOLECULAR  MAGNITUDES  283 

spheres.  But  as  we  shall  see  in  considering  the  motion 
of  the  helium  ions,  which  are  shot  off  from  radium, 
the  molecule  is  more  like  a  solar  system  with  relatively 
large  spaces  between  the  electrons  and  the  nucleus 
which  compose  it.  That  matter  is  composed  of  finite 
particles  which  are  in  constant  motion,  and  that 
phenomena  like  gaseous  pressure,  osmotic  pressure, 
and  viscosity  are  due  to  motions  and  impacts  of  these 
small  particles  is  verified  by  this  remarkable  evidence. 

Two  practical  applications  of  the  fundamental  facts 
of  the  kinetics  of  gases  are  illustrated  in  the  pumps  for 
obtaining  high  vacua  in  the  manufacture  of  vacuum 
devices  such  as  X-ray  tubes  and  thermionic  vacuum 
tubes  for  radio-communication.  The  vacua  to  be 
obtained  to-day  are  far  in  excess  of 
those  obtained  by  Torricelli.1 

Consider  for  example  a  rotating 
drum.  Molecules  of  the  adjacent  gas, 
which  impinge  upon  it,  have  super- 
imposed upon  their  natural  molecular 
motions  a  mass  motion  in  a  direction 
tangential  to  the  drum.  If  the  mean 
free  path  is  small  they  will  lose  this  FlG-  35- 
forced  motion  by  collisions  with  other  molecules,  but 
if  it  is  comparatively  large  they  will  maintain  it  during 
an  appreciable  travel.  In  the  Gaede  molecular  pump 
the  drum  rotates  inside  a  hollow  cylinder,  as  shown 
in  cross  section  in  Fig.  35. 

The  vessel  to  be  exhausted  is  connected  at  A.     An 

1  Above  the  mercury  in  his  barometer  there  was,  of  course,  the 
vapor  of  the  mercury.  In  accurate  readings  of  a  mercury  barometer 
an  allowance  is  always  made  for  this  fact. 


284         THE   REALITIES  OF   MODERN  SCIENCE 

ordinary  air  pump,  connected  to  B,  maintains  the 
pressure  within  the  cylinder  low  enough  to  permit  the 
desired  action.  Molecules,  due  to  their  natural  motions, 
enter  C  from  A  and  impinge  on  the  drum  D.  They 
leave  D  with  a  component  motion  in  the  tangential 
direction  of  the  small  arrow  a.  Because  the  mean  free 
path  is  large  few  of  them  encounter  other  molecules. 
Striking  against  the  walls  of  C,  they  are  reflected  back 
on  to  the  drum  but  farther  along.  In  this  way  they 
are  impelled  around  C  from  A  to  p.  The  number  of 
molecules  at  the  exit  B  is  therefore  maintained  greater 
than  at  A.  The  action  of  the  " rough"  pump  con- 
stantly removes  some  of  these  excess  molecules. 

With  such  a  molecular  pump1  vacua  corresponding 
to  2X10~8  cm.  of  Hg  have  been  obtained.  This 
represents  a  pressure  of  about  three  ten-thousandths 
of  a  dyne/cm.2  In  oxygen  the  mean  free  path  cor- 
responding to  this  pressure  is  76/2  X10~8  times  the 
value  of  page  279,  or  35,400  cm.  The  number  of  mole- 
cules per  c.  c.  is  directly  as  the  pressure  and  is  ap- 
proximately 7X109.  The  number  of  collisions  per 
second  per  molecule  is  the  quotient  of  the  average 
velocity,  which  has  not  been  altered  by  a  reduction 
of  the  pressure,  and  the  mean  free  path,  or  about  one 

1  In  the  actual  pump  there  is  a  series  of  chambers  like  C.  The 
first  of  these  connects  at  A  to  the  vessel  to  be  exhausted  and  the 
last  connects  at  B  to  the  backing  pump.  The  outlet  B  of  the  first 
chamber  connects  at  B  with  the  intake  A  of  the  succeeding  chamber, 
and  so  on.  The  result  is  equivalent  to  that  which  would  be  pro- 
duced by  a  drum  of  much  larger  diameter.  The  molecules  which 
are  impelled  around  toward  B  have,  therefore,  smaller  chance  of 
diffusing  back  to  A.  The  actual  pump  runs  in  oil  so  arranged  that 
by  the  rotation  the  oil  is  thrown  outward  against  the  clearance 
spaces  at  the  end  of  the  drums  and  thus  seals  them  very  effectively. 


MOLECULAR   MAGNITUDES 


285 


collision  per  second.  This  gives  an  interesting  picture 
of  the  spatial  relations  of  the  molecules  in  a  c.  c.  of 
highly  rarefied  gas.  Although  there  are  about  seven 
thousand  million,  they  are  so  widely  separated,  rela- 
tive to  their  size,  that  each  molecule  travels  on  the 
average  about  a  third  of  a  kilometer  between  col- 
lisions and  makes  but  one  collision  a  second  with  an- 
other gas  molecule. 

The  principle  of  the  other  pump  is  somewhat  similar, 
but  the  mass  motion  of  the  molecules  which  are  to  be 
exhausted  is  due  to 
impacts  with  directed 
molecules.  The  sys- 
tem is  shown  in  Fig. 
36.  The  rough  pump 
is  connected  at  B  and 
the  vessel,  which  is 
to  be  evacuated,  at  A. 
The  reservoir  R  con- 
tains mercury  which 
is  heated  and  boils. 
The  mercury  molecules  rise  in  a  dense  stream  and  shoot 
through  the  tube  c  into  the  chamber  C.  Molecules  of 
the  gas  entering  at  A  get  in  the  way  of  this  stream, 
and  receive  a  mass  motion  toward  the  outlet  at  B. 
The  chamber  C  is  cooled,  usually  by  a  water  jacket 
(not  shown  in  the  sketch),  and  hence  the  vapor  mole- 
cules condense  on  its  sides  and  trickle  down  again  into 
the  reservoir  R. 

The  kinetic  energy  of  the  mercury  vapor  molecules  is 
in  part  transferred  to  the  gas  molecules  with  which 
they  happen  to  come  into  collision,  but  in  greater  part 


FIG.  36. 


286         THE   REALITIES  OF   MODERN  SCIENCE 

is  removed  by  this  cooling.  At  the  upper  end  of  the 
chamber  C  there  are  some  molecules  of  the  vapor,  but 
these  are  much  reduced  in  velocity  by  the  cooling  and 
but  few  of  them  find  their  way  back,  against  the  stream, 
into  the  region  A.  The  stream  of  molecules  from  c 
spreads  out  somewhat  as  a  result  of  the  haphazard 
motions  of  its  individual  molecules.  The  mass  motion 
given  to  the  gas  molecules  may  therefore  be  lateral  in 
part,  in  which  case  they  may  bounce  a  few  times  from 
the  walls  of  C,  but  the  motion  is  nevertheless  in  the 
general  direction  of  the  exit,  B.  The  lateral  spreading 
of  the  vapor  molecules  means  collisions  with  the  walls 
and  hence  a  transfer  of  part  of  their  energy  to  the 
cooler  molecules  of  the  walls.  In  this  way  but  few 
of  them  reach  the  exit  C. 

In  commercial  form  this  pump  has  been  developed 
along  lines  proposed  by  Langmuir.  The  limit  of  the 
vacua  to  be  obtained  by  its  use  does  not  appear  to 
be  inherent  in  the  pump  itself  but  to  depend  upon  the 
character  of  the  walls  of  the  vessels,  particularly  that 
which  is  to  be  exhausted.  We  recognize  that  these 
walls  are  not  smooth  and  solid  but  are  a  granular 
structure  of  glass  molecules.  Within  the  spaces  be- 
tween these  molecules  there  may  be  numbers  of  gas 
molecules.  These  diffuse  into  what  we  usually  con- 
sider the  free  space  of  the  vessel,  but  do  so  compara- 
tively slowly  because  of  the  network  of  glass  molecules 
past  which  they  must  thread  their  way.  Gas,  which 
is  thus  occluded  by  the  walls  of  the  vessel,  is  probably 
the  real  limit  of  the  vacuum  which  may  be  obtained, 
for  otherwise,  sooner  or  later,  every  gas  molecule  of 
the  vessel  at  A  would  find  its  way  into  the  chamber  C. 


MOLECULAR  MAGNITUDES  287 

Modern  methods  which  are  used  for  measuring 
small  gaseous  pressures  also  illustrate  the  reality  and 
the  fundamental  facts  of  molecular  kinetics.  A  typical 
pressure  gauge  consists  of  a  chamber  which  is  connected 
to  the  vessel  containing  gas  at  the  unknown  pressure. 
It  contains  a  small  strip  of  platinum  and  a  movable 
vane.  The  strip  is  heated  electrically.  The  de- 
flection of  the  vane  is  manifested  by  the  rotation  of 
a  beam  of  light  reflected  from  it.  The  vane  turns  as 
a  result  of  its  bombardment  by  molecules  which  have 
been  heated  by  contact  with  the  platinum  strip.  The 
pressure  at  the  vane  is  therefore  higher  than  elsewhere 
in  the  vessel.  The  uniform  pressure  in  the  rest  of  the 
vessel  may  be  expressed  in  terms  of  the  pressure  on  the 
vane  and  the  temperatures  of  the  vane  and  the  plati- 
num strip.  With  a  gauge  of  this  type  pressures  have 
been  observed  as  small  as  5X10~10  cm.  of  Hg  (i.e.  7 
millionths  of  a  dyne  per  sq.  cm.). 


CHAPTER  XXI 
MOLECULAR  ENERGY 

THE  value  of  the  kinetic  energy  of  translation  of  all 
the  molecules  in  a  mole  has  been  shown  to  be  3RT/2. 
To  raise  the  temperature  of  a  mole  one  degree  therefore 
requires  3R/2  ergs  to  supply  the  increases  in  energy  of 
translation  of  the  molecules,  that  is,  12/2  ergs  for  each 
degree  of  freedom.  (If  the  molecule  has  other  de- 
grees of  freedom  additional  energy  will  be  required.) 
If  expansion  is  allowed  and  external  work  done  by  the 
gas,  still  more  energy  is  required  to  heat  the  mole  one 
degree.  Represent  this  energy  by  A. 

In  considering  the  amount  of  energy  required  per 
degree  of  temperature  per  mole  we  therefore  distin- 
guish two  cases,  namely,  (1)  the  energy  required  when 
the  volume  is  maintained  constant,  and  (2)  that  re- 
quired when  the  volume  is  allowed  to  increase  but  the 
external  pressure  is  maintained  constant.  The  first 
of  these  is  the  "  specific  heat  at  constant  volume," 
denoted  by  Cv,  and  the  second  is  the  "specific  heat  at 
constant  pressure,"  Cp.  We  may  write  an  expression 
for  A,  the  external  work  per  mole  per  degree  of  tem- 
perature, and  these  specific  heats  as 

CP=CV+A  (1) 

These  magnitudes  are  conventionally  expressed  in 
other  units  than  the  erg.  The  usual  unit  is  the  calorie, 

288 


MOLECULAR  ENERGY  289 

which  is  equal  to  an  odd  multiple  of  an  erg,  namely, 
4.19X107,  or  4.19  joules.  The  calorie  is  a  relic  from 
the  days  of  the  phlogiston  theory,  which  has  become 
firmly  fixed  in  the  literature. 

It  was  adopted  at  a  time  when  heat  appeared  or  dis- 
appeared from  the  view  of  experimenters  in  a  most 
mysterious  manner,  and  was  a  measure  of  that  "  im- 
ponderable fluid."  It  was  and  is,  however,  a  very 
convenient  unit  in  many  ways.  For  example,  it  was 
found  that  if  two  masses  of  water,  mi  and  7^2,  at  tem- 
peratures of  h  and  iz  respectively,  are  mixed  the  mix- 
ture comes  to  a  temperature,  t,  given  by  the  relation 1 

m1(tl-t)=m2(t-t^)  (2) 

It  is  convenient  to  take  unit  heat  as  that  lost  or  gained 
when  1  gram  of  water  changes  1°  C.  This  is  the  calorie. 
In  calories,  then,  the  " specific  heat"  of  water  is  unity. 
We  now  see  that  what  Joule  (cf.  page  112)  really  meas- 
ured was  the  specific  heat  of  water.  Up  to  his  time 
its  absolute  value  was  unknown.  The  calorie  is  merely 
an  arbitrarily  chosen  unit  for  measuring  energy.  It 
is  therefore  usual  to  write  1  calorie  =  J  ergs,  where  J 
is  sometimes  incorrectly  called  the  "  mechanical  equiv- 
alent of  heat." 

This  " method  of  mixture"  which  led  to  the  adoption 
of  the  calorie  as  a  unit  of  "heat"  was  easily  extended. 

1  This  equation  means  that  the  molecules  of  the  mass  mi,  which 
have  a  kinetic  energy  of  translation  corresponding  to  ti,  arrive  at 
an  equilibrium  with  the  molecules  of  mass  ra2,  which  are  at  the 
lower  temperature  t?,  when  both  groups  of  molecules  have  the  same 
average  kinetic  energy  of  translation  and  hence  a  common  tem- 
perature t.  The  energy  transferred  from  the  molecules  of  the  first 
mass  is  m\(ti—t)  and  in  a  conservative  system  equals  that  gained 
by  the  other  molecules,  namely 


290         THE   REALITIES   OF   MODERN   SCIENCE 

For  example,  if  a  mass  of  some  substance  other  than 
water  was  heated  and  mixed  with  a  mass  of  water  at 
a  lower  temperature,  the  final  temperature  was  found 
to  be  lower  than  equation  (2)  indicated.  The  sub- 
stance was,  therefore,  said  to  have  a  smaller  capacity 
for  the  intangible  fluid.  For  calculations  the  mass 
of  the  substance  was  therefore  reduced  to  that  of  an 
amount  of  water,  equivalent  in  heat  capacity,  by 
multiplying  it  by  a  factor  called  its  specific  heat. 
The  general  relation  of  which  equation  (2)  is  a  special 
case  is  then 

MlS1(t1-t)=MA(t-ti)  (3) 

where  Si  and  S2  are  the  specific  heats  1  in  calories  per 
gram. 

It  is  convenient  in  discussing  gases  to  deal  with  the 
molecular  specific  heat,  that  is,  the  heat  per  degree 
per  mole,  as  we  have  in  equation  (1).  Let  us  retain 
the  same  symbols  but  express  Cv  and  Cp  in  calories; 
since  A  equals  R,2  we  have 

CP=CV+R/J  (4) 

1  The  calorists  found  a  curious  state  of  affairs  when  one  of  the 
components  of  the  mixture  passed  through  a  change  of  state.     For 
example  one  gram  of  steam  at  100°  C.  in  changing  from  steam  to 
water  without  change  of  temperature  would  liberate  about  536 
calories.     To-day  we  recognize  in  this  change  a  conversion  of  poten- 
tial energy  into  kinetic.     The  calorists,  however,  considered  that 
there  was  a  hidden  store  of  the  fluid  which  was  released  when  steam 
condensed  and  called  the  amount  per  gram  the  "latent  heat  of 
vaporization."     Similarly  when  water  freezes  about  80    calories 
per  gram  are  released  merely  by  the  change  of  state.     This  they 
called  the  "latent  heat  of  fusion."     These  two  unfortunate  terms 
remain  in  the  literature  to-day. 

2  Consider  1  mole,  at  pressure  p  and  temperature  T,  to  be  haated 
1°  Cr  and  to  do  external  work,  A,  against  a  piston  of  area,  a.    Lot 


MOLECULAR  ENERGY  291 

Let  us  now  see  what  value  of  Cv  we  should  expect 
for  various  types  of  molecules.  In  addition  to  trans- 
lation a  monatomic  molecule  should  have  three  degrees 
of  rotation,  that  is  of  spin,  about  three  mutually  per- 
pendicular axes  through  its  center.  With  the  fact 
that  a  body  has  such  degrees  of  freedom  we  are  familiar 
from  our  experience  with  golf  balls.1  Unless  the  centers 
of  two  molecules  are  moving  along  the  same  line 
or  unless  they  are  perfectly  smooth  spheres,  a  collision 
results  in  a  change  in  their  rotation.  In  an  impact 
the  points  of  contact  come  to  rest  relatively.  The 
centers,  and  in  fact  all  other  points,  do  not,  but  con- 
tinue in  their  original  directions  because  of  their 
inertias.  If  the  impact  is  elastic  the  molecules  bounce 
apart,  but  their  rotational  speeds  will  have  been  altered. 

The  k.e.  of  rotation  of  the  various  molecules  of  a 
gas  will,  therefore,  be  different  and  that  of  the  same 
molecule  will  vary  from  time  to  time.  It  will,  how- 
ever, have  an  average  value  just  as  in  the  case  of  trans- 
lation. According  to  the  conclusions  of  the  mathema- 
ticians the  two  average  values  will  be  equal,  that  is, 
there  will  be  equipartition  between  translation  and 
rotation  and  hence  between  all  the  degrees  of  freedom. 

Corresponding  to  an  increase  of  1°  C.  in  a  mole  of 
gas  each  degree  of  freedom  of  translation  is  increased 
by  R/2  ergs.  From  page  218  we  have  #  =  83.2  XlO6 

the  initial  and  final  volume  be  Vi  and  F2,  respectively.  The 
pistonmoves  (F2-  Fi)/a  against  aforce,  pa,  and  hence  A  =  p(V2—  Vi). 
But  pVi=RT  and  pV2  =  R(T+l),  hence  A  =  R. 

1  In  this  case  we  usually  recognize  the  existence  of  a  spin  by  a 
deflection  of  the  ball  from  its  rectilinear  path  due  to  the  difference 
in  air  pressure  on  the  two  sides,  which  is  occasioned  by  its  rotation 
in  combination  with  its  translation. 


292         THE   REALITIES  OF   MODERN  SCIENCE 

ergs,  and  hence  R/2J  is  0.99  calorie.  For  conven- 
ience *  in  discussion  we  shall  use  the  approximate 
value  of  1  calorie.  We  then  expect  the  specific  heat, 
Cv,  to  be  (3+n)E/2J  or  (3+n)  calories,  where  n  is 
the  number  of  degrees  of  freedom  which  the  molecule 
has  in  excess  of  the  three  of  translation.  For  a  mona- 
tomic  molecule,  therefore,  we  put  n  =  3  and  expect 
Cv  to  be  (3+n)  or  6  calories. 

Specific  heat,  we  notice,  is  merely  the  name  for  a 
rate,  that  of  change  of  the  total  molecular  energy  per 
degree  of  temperature.  We  expect  this  rate  for  a 
monatomic  molecule  to  be  6  calories,  independent 
of  the  temperature.  Now,  what  are  the  facts?  In 
the  first  place,  we  find  that  specific  heats  are  not  in 
general  independent  of  the  temperature,  as  we  shall 
see  in  more  detail  later.  In  the  second  place,  we  find 
that  the  specific  heat  of  monatomic  gases  is  practically 
3  instead  of  6  calories.  For  argon  it  has  been  measured 
over  a  very  wide  range  of  temperature  and  found  to  be 
constant  and  of  value  2.98  cal.  Apparently  all  the 
energy  required  to  raise  a  mole  of  argon  1°  C.  is  that 
needed  to  increase  its  kinetic  energy  of  translation. 
According,  however,  to  the  simple  mechanical  con- 
siderations of  the  last  few  pages  any  impacts,  except 
head-on,  must  cause  some  rotation  of  the  molecules 
unless  the  latter  happen  to  be  smooth  spheres.  Of 
course,  if  the  argon  atom  was  practically  a  geometrical 
point,  that  is,  infinitely  small,  it  could  not  rotate.  But 

1  This  convenience  may  not  be  credited  to  the  calorists.  In  this 
connection  it  is  interesting  to  note  that  before  Joule's  classical  ex- 
periments on  the  "mechanical  equivalent  of  heat,"  J.  R.  Mayer  had 
reasoned  from  the  relation,  CP  —  CV=  external  work,  and  had  ob- 
tained a  value  of  the  calorie  in  mechanical  units. 


MOLECULAR  ENERGY  293 

we  know  that,  although  it  is  small,  it  is  finite  in  diam- 
eter. Helium  and  mercury  vapor,  which  are  both 
monatomic  gases,  also  have  values  of  molecular  heat 
of  about  3  calories  and  behave  as  if  there  was  no  energy 
of  rotation. 

In  the  case  of  a  diatomic  gas  there  are  five  degrees 
of  freedom,  if  the  distance  between  the  centers  of  the 
atoms  remains  fixed.  The  position  of  one  atom  is 
then  determined  with  reference  to  the  other  by  two 
angles  which  are  formed  by  the  connecting  line  and 
two  reference  planes,  one  angle  giving  the  bearing  or 
" azimuth"  and  the  other  the  height  or  " elevation"  as 
measured  from  a  plane  perpendicular  to  the  first 
reference  plane.  Such  a  system,  therefore,  has  five 
degrees  of  freedom,  and  we  should  expect  its  specific 
heat  to  be  5  calories. 

Many  of  the  diatomic  gases  have  molecular  heats 
of  about  this  value.  For  example,  oxygen  has  the 
following  values:  Cv  is  5.17  at  300°  C.,  5.35  at  500°, 
and  6.00  at  2000°  C.  It  will  be  noticed  that  although 
in  one  instance  Cv  has  about  the  expected  value,  it 
does  vary  and  is  higher  at  the  higher  temperatures. 
At  the  higher  temperatures  oxygen  seems  to  be  more 
nearly  in  the  condition  of  the  gases  which  we  know 
dissociate.  For  such  gases  we  find  the  value  of  the 
molecular  heat  about  6  at  ordinary  temperatures  and 
higher  than  that  at  still  higher  temperatures.  It 
seems  as  though  the  bonds  which  maintain  the  dis- 
tance between  the  two  atoms  were  gradually  weakening, 
allowing  an  oscillation  of  the  atoms  in  the  case  of  more 
and  more  molecules  as  higher  temperatures  are  reached. 

If  there  is  a  motion  of  one  atom  relative  to  the 


294         THE   REALITIES   OF   MODERN   SCIENCE 

other  there  are  six  degrees  of  freedom,  namely,  three  of 
translation,  two  of  rotation,  and  one  of  oscillation. 
In  an  oscillation,  however,  there  is  a  continuous  change 
of  kinetic  energy  into  potential  and  then  of  potential 
into  kinetic.  According  to  the  generalization  of  the 
equipartition  of  energy  the  kinetic  energy  of  this 
oscillation  is  on  the  average  1  calorie.  Now  the 
average  value  of  the  potential  energy  must  be  the 
same.1  There  should  then  be  required  an  average  of 
1  calorie  of  kinetic  and  1  calorie  of  potential  energy, 
or  a  total  of  two  calories  for  this  degree  of  freedom. 
The  molecular  heat  should  then  be  7  calories  and  not 
6.  We,  therefore,  expect  the  molecular  heat  of  a  dia- 
tomic molecule  to  be  either  5  or  7  calories,  depending 
upon  whether  or  not  there  is  freedom  of  oscillation, 
but  we  do  not  expect  intermediate  values.  Such, 
however,  are  found  in  actual  experiment. 

The  idea  of  an  equipartition  of  energy  among  the 
degrees  of  translation  was  seen  in  Chapter  XIII  to  lead 
to  a  correct  expression  of  the  kinetic  relations  for  gas 
molecules.  An  equipartition  between  translation  and 
rotation  (or  internal  oscillations)  would  seem  from 
page  291  to  be  physically  necessary  for  granular  atoms 
composed  of  electrons.  As  a  matter  of  fact,  one  of  the 
methods  employed  by  Perrin  in  determining  the  value 

1  For  convenience  in  this  relative  motion  consider  that  one  atom 
oscillates  about  a  mean  position  with  reference  to  the  other.  The 
oscillating  atom  has  its  maximum  k.e.  while  passing  through  this 
unstressed  position.  At  the  turning  points,  both  nearer  and  farther 
from  the  other  atom,  there  is  zero  k.e.,  but  in  these  positions  it  has 
a  potential  energy  equal  to  its  maximum  k.e.  The  result  is  that  the 
k.e.  varies  from  a  maximum  to  zero  while  the  p.e.  is  varying  from 
zero  to  an  equal  maximum.  The  average  values  of  the  p.e.  and  the 
k.e.  are  then  equal. 


MOLECULAR  ENERGY  295 

of  N  involved  Brownian  rotations  1  of  spherules  of 
mastic.  The  equation  which  he  used  to  interpret 
his  observations  was  correct  only  if  there  was  on  the 
average  an  equipartition  of  energy  between  rotation 
and  translation  for  these  spheres.  Of  course,  the  latter 
had  to  be  large  enough  so  that  he  could  observe  any 
markings  on  them  and  thus  detect  and  measure  the 
angular  displacements  of  such  markings.  He  used 
spheres  of  diameter  13/*  (13X10~3mm.),  which  were 
therefore  about  100,000  times  as  large  in  diameter  as 
a  molecule.  For  particles  at  least  as  small  as  this, 
therefore,  equipartition  holds. 

From  the  values  of  the  specific  heat  of  monatomic 
gases  it  appears  that  the  degrees  of  freedom  of  rota- 
tion, which  we  should  expect  the  molecules  to  have, 
are  effectively  ankylosed 2  at  all  temperatures  to  which 
they  have  yet  been  subjected.  It  further  appears 
from  the  values  for  other  gases  that  as  the  temperature 
increases  there  is  an  increasing  number  of  molecules 
for  which  there  is  no  longer  ankylosis.  There  is  also 
reason  to  believe  that  collisions  of  the  molecules 
result  in  the  acquisition  of  energy  in  these  degrees  of 
freedom  only  if  a  definite  quantity  of  energy  is  made 
available  by  the  impact.  If,  at  the  temperature  rep- 
resented by  the  average  molecule,  the  velocity  of 
translation  of  the  fastest  molecules  is  too  small  for 
a  collision  to  result  in  this  quantum  of  energy,  then 
none  of  the  k.e.  of  translation  is  absorbed  by  other 
degrees  of  freedom. 

1  Cf.  Table  II,  p.  282. 

2  The   term    "ankylosis"    entered   the   language   of   statistical 
mechanics  from  medicine,  where  it  represents  a  stiffening  or  fixation 
of  a  joint. 


296         THE   REALITIES  OF   MODERN   SCIENCE 

The  more  thoroughly  one  is  accustomed  to  the  scien- 
tific formulation  of  the  laws  of  matter  and  energy 
which  hold  for  visible  masses,  the  more  impossible 
seems  such  a  phenomenon  as  has  been  suggested.  We 
expect  that  a  system  will  absorb  energy  in  any  amount 
and  increase  continuously  in  its  energy,  and  we  cannot 
as  yet  visualize  a  molecular  mechanism  of  the  sort  we 
have  implied  above. 

In  the  case  of  solids  the  principle  of  equipartition  is 
even  farther  from  representing  the  facts.  The  mole- 
cule of  a  solid  cannot  leave  its  boundaries  and  must 
have  all  its  energy  in  oscillation  about  a  mean  position. 
As  it  departs  from  that  position  it  forms  with  adjacent 
molecules  systems  of  increasing  potential  energy.  As 
it  returns  toward  the  mean  position  this  potential 
energy  is  converted  into  kinetic  energy.  Heating  the 
solid  increases  the  kinetic  energy  of  the  "  oscillators," 
as  we  might  call  them.  The  average  potential  energy, 
however,  increases  equally  with  the  kinetic.  There 
are  three  mutually  perpendicular  axes  along  which 
the  molecule  may  vibrate  and  there  will  be  required 
one  calorie  of  p.e.  and  one  of  k. e.  for  each  of  these 
three  degrees  of  vibrational  freedom.  We  should 
therefore  expect  a  total  of  6  calories. 

The  molecular  heat  of  monatomic  solids,  that  is,  the 
atomic  heat  of  elementary  substances,  is  in  many 
instances  about  6  calories,  as  is  evident  from  the  follow- 
ing values:  carbon  5.5,  boron  5.5,  sodium  6.7,  magne- 
sium 5.9,  phosphorus  6.2,  sulphur  5.9,  silver  6.0, 
copper  5.9,  iron  6.4,  nickel  6.3,  aluminium  5.9,  zinc 
6.1,  platinum  6.2,  gold  6.2,  cobalt  6.0.  In  the  early 
days  of  calorimetric  measurements  (1818)  Dulong  and 


MOLECULAR  ENERGY  297 

Petit  noticed  the  fact  that  the  product  of  the  specific 
heat  per  gram  and  the  atomic  weight  was  approxi- 
mately 6.4  calories.  They  announced  the  fact  as  an 
empirical  law,  years  before  the  idea  of  an  equiparti- 
tion  had  been  developed. 

More  recent  measurements  have  shown  that  this 
relationship  is  the  accidental  result  of  the  temperatures 
of  their  experiments,  for  as  the  temperature  is  decreased 
the  specific  heat  is  found  to  have  lower  values,  approach- 
ing zero  at  the  absolute  temperature  of  zero.  Thus 
Cv  for  carbon  (from  measurements  on  a  diamond)  is 
5.51  at  1258°  absolute;  5.29  at  880°;  3.63  at  520°; 
2.12  at  358° ;  1.35  at  283°  (10°  C.) ;  0.76  at  222°, 
and  practically  zero  at  50°  absolute. 

As  a  solid  is  heated,  its  changes  in  temperature  are 
manifested  to  our  senses  by  the  radiation  of  heat,  and 
later  by  luminous  radiations,  first  dull  red,  but  finally 
all  those  of  the  spectrum,  hi  which  condition  the  solid 
is  incandescent.  According  to  the  modern  theory,  this 
light  is  due  to  the  oscillations  of  the  electrons.  These 
electronic  oscillators  we  should  expect  to  be  displaced 
from  their  equilibrium  positions  by  collisions  or  by 
interactions  with  the  electrons  of  atoms  in  their  im- 
mediate neighborhood.  According,  however,  to  the 
theory  advanced  by  Planck,  they  will  be  caused  to  oscil- 
late only  by  the  receipt  of  a  definite  amount  of  energy, 
a  quantum.  The  quantum  is  not  necessarily  the  same 
for  two  different  oscillators,  but  depends  upon  their 
natural !  frequencies,  being  d  =  hf,  where  /  is  the  fre- 
quency and  h  is  known  as  "  Planck's  constant. " 

1  The  natural  frequency  of  a  system  is  the  frequency  at  which  it 
will  oscillate  if  displaced  from  its  equilibrium,  and  then  left  free 


298         THE   REALITIES   OF   MODERN   SCIENCE 

The  total  energy  possessed  by  the  oscillators  of  one 
mole  of  a  substance  will  then  depend  upon  their 
natural  frequencies,  upon  the  number  of  molecules 
in  the  mole,  A7",  and  upon  the  absolute  temperature,  T. 
As  the  temperature  is  changed  the  energy  changes 
and  the  ratio  of  the  change  in  energy  to  the  change  in 
temperature  is,  of  course,  the  specific  heat.  In  terms 
of  h,  /,  N,  and  T  both  the  average  energy  and  the 
specific  heat l  have  been  expressed  by  Einstein.  These 
formulae  do  not  give  calculated  values  of  the  specific 
heat  exactly  in  accord  with  the  experimental  values, 
although  the  agreement  is  sufficient  over  a  wide  range 
of  temperatures  to  indicate  that  in  the  concepts  of 
energy  quanta  and  of  electronic  oscillators,  scientists 
have  much  more  than  a  hopeful  clew  to  follow  in  their 
study  of  matter  and  energy.  Some  modifications  of 
Einstein's  equation  have  been  suggested  which  adapt 
it  more  nearly  to  the  observed  facts  as  to  the  specific 
heat  of  solids,  but  these  we  shall  not  consider. 

To  apply  such  a  formula  it  is  necessary  to  have 
numerical  values  for  h  and  /.  The  value  of  Planck's 
constant,  h,  has  been  determined  several  times  and  is 
fairly  accurately  known,  as  we  shall  see  in  the  suc- 
ceeding chapter.  As  to  the  frequency,  /,  of  the  natural 
oscillation  it  should  be  noted  that  in  general  there  would 
be  more  than  one  natural  frequency.  (The  total 

of  external  influences  to  dissipate  the  energy  contributed  to  it  in 
producing  the  displacement.  The  opposite  of  a  "  natural ' '  vibration 
is  a  "forced"  one.  A  violin  string  when  bowed  receives  energy  and 
vibrates  at  its  natural  frequency,  but  the  drum  of  the  ear  of  the 
listener  undergoes  a  forced  vibration. 
1Thn.  r  -O 

c'-3 


MOLECULAR   ENERGY  299 

specific  heat  would  then  be  the  sum  of  the  specific 
heats  as  found  for  each  of  the  types  of  oscillators.)  The 
determination  of  these  frequencies  may  be  made  by 
several  methods,  of  which  two  will  be  described. 

It  is  found  that  substances  absorb  the  same  fre- 
quencies as  they  naturally  emit.  Thus  glass  which 
transmits  red  light  absorbs  blue ;  when  heated  it  will 
be  found  to  give  off  a  blue  light,  not  a  red  light.  The 
frequencies  which  are  reflected  from  the  surface  are 
found  to  be  those  which  are  most  strongly  absorbed 
in  transmission.  Thus  for  this  case,  blue  light  will  be 
reflected  from  the  outer  surface. 

To  determine  what  frequencies  are  reflected  from  a 
substance  it  is  then  only  necessary  to  illuminate  it  by 
a  source  containing  all  frequencies  1  and  to  observe 
the  reflected  light.  Now,  at  first  thought,  this  does 
not  seem  reasonable,  for  we  know  if  we  observe  the 
image  of  an  incandescent  electric  lamp  as  reflected  by 
a  sheet  of  red  glass  that  the  lamp  appears  white, 
exactly  as  well  as  if  viewed  directly.  In  this  particular 
case  there  happens  to  be  reflected  from  the  polished 
surface  enough  light  of  all  the  frequencies  which  com- 
pose white  light  so  that  we  do  not  observe  the  expected 
effect.  But  let  us  arrange  other  mirrors  of  red  glass 
and  see  the  lamp  by  reflection  from  several  surfaces 
instead  of  only  one.  Each  successive  reflection  re- 
sults in  purer  and  purer  light  "of  the  natural  fre- 
quencies of  the  red  glass. "  The  rays  of  light  which 
survive  a  number  of  such  reflections  are  called  "re- 

*A  "full  radiator,"  containing  all  frequencies  from  zero  to  in- 
finity, sometimes  called  a  "black  body,"  since  the  ideal  black  body 
would  absorb  all  frequencies. 


300         THE   REALITIES   OF   MODERN  SCIENCE 

sidual  rays."  It  is  not  correct  to  say  "of  the  natural 
colors/ '  for  some  of  the  "rays"  may  be  of  frequencies 
too  low  to  be  visible  and  others  too  high. 

The  phenomenon  of  residual  rays  is  one  of  "reso- 
nance," of  which  many  illustrations  are  to  be  found. 
Consider  a  case  in  sound.  If  all  the  keys  of  a  piano 
are  depressed  so  that  the  strings  are  free  to  vibrate 
and  if  then  a  person  sings  some  note,  the  string  cor- 
responding to  this  note  is  set  into  "sympathetic 
vibration."  The  energy  of  the  sound  waves  is  nob 
absorbed  by  the  other  strings  to  an  appreciable  extent, 
although  they  are  all  forced  to  move  somewhat.  In 
the  case  of  the  string  of  the  proper  natural  frequency, 
each  succeeding  displacement  which  the  wave  causes 
is  in  just  the  same  direction  as  the  string  would  natu- 
rally move  as  a  result  of  the  energy  imparted  during 
the  preceding  displacement.  The  energy  of  the  wave 
train  is,  therefore,  always  available  for  increasing  the 
displacement  of  the  string.  In  other  cases  this  energy 
is  obviously  not  always  so  available.  The  greater  the 
musical  interval  between  the  natural  frequency  of 
the  string  and  the  impressed  frequency  of  the  wave 
train,  the  more  nearly  will  it  be  true  that  energy  is 
available  alternately,  and  in  equal  amounts,  for  reduc- 
ing the  natural  motion  of  the  string  and  for  increasing 
it.  Under  these  conditions  the  string  will  have  no 
motion. 

Looking  at  this  matter  from  a  slightly  different 
viewpoint,  we  see  that  this  means  that  the  string  of 
the  wrong  frequency  does  not  absorb  energy  from  the 
wave  train.  All  the  energy  of  the  wave  train  passes 
by  such  strings.  We,  therefore,  do  not  expect  a 


MOLECULAR   ENERGY  301 

medium  to  transmit  a  wave  train  without  absorption 
when  the  natural  frequency  of  its  oscillators  is  the  same 
as  that  of  the  waves.  Those  oscillators  of  a  medium 
which  are  nearest  the  source  of  the  radiation  absorb 
the  most  and  those  farthest  away  the  least  of  the  energy. 
For  this  reason  a  thin  sheet  of  black  paper  may  prove 
insufficient  to  protect  photographic  films  in  strong  sun- 
light when  a  sheet  of  double  the  thickness  may  do  so. 

The  oscillators  of  a  medium  which  are  absorbing 
radiation  are  vibrating  with  the  increased  amplitudes. 
Now,  it  makes  no  difference  how  they  were  set  into 
vibration,  they  will  also  radiate  energy.  (It  is  im- 
material, for  example,  whether  a  piano  string  is  set 
into  vibration  by  a  wave  train  or  by  a  blow  on  the 
proper  key,  it  will  send  out  a  wave  tram  of  its  own 
frequency.)  The  absorbing  oscillators,  at  and  near 
the  surface,  thus  act  as  radiators.1  The  frequency  of 
the  radiation,  which  is  " reflected,"  must  then  be  that 
which  is  absorbed.  Reflection  is  really  " re-radiation." 

In  the  other  method  for  obtaining  the  natural  fre- 
quencies it  is  assumed  that  at  the  melting  point  of  a 
solid  the  amplitude  of  vibration  of  the  atoms  is  prac- 
tically equal  to  the  average  distance  between  the  centers 
of  adjacent  molecules.  At  the  melting  point  of  all 

1  The  reason  that  a  mirror  surface  must  be  plane  and  smoothly 
polished  is  now  apparent.  It  must  present  to  the  impressed  radi- 
ation a  smooth  front  of  oscillators  so  that  the  energy  which  they 
radiate  may  all  be  available  at  some  distant  point.  If  the  surface 
is  irregular  the  energy  radiated  by  some  of  the  oscillators  will  render 
unavailable  at  the  desired  distant  point  some  of  the  energy  radiated 
by  the  other  oscillators.  Or,  as  it  is  more  usually  said,  there  will 
be  a  scattering  and  an  interference  of  the  reflected  waves.  Scatter- 
ing is  negligible  if  the  irregularities  are  small  as  compared  to  the 
wave  length  of  the  incident  radiation. 


302         THE   REALITIES  OF   MODERN   SCIENCE 

solids  the  atomic  heat  is  practically  6  calories,  that  is, 
Dulong  and  Petit's  law  holds  for  these  high  tempera- 
tures. (It  might  be  noted  that  we  should  expect  the 
physical  behavior  of  molecules  to  be  alike  at  "  cor- 
responding states.")  An  expression  may  then  be 
written  for  the  frequency  in  terms  of  known  constants 
of  the  substance  under  examination.  The  values 
obtained  by  this  method  are  not  sufficiently  accurate, 
however,  to  warrant  their  use  in  calculations,  but 
they  do  serve  as  checks  on  those  obtained  by  other 
methods. 

In  the  matter  of  electronic  oscillators  and  their 
ability  to  absorb  energy  we  have  reached  a  pivotal 
position  of  modern  science.  The  earlier  idea  as  to 
the  equipartition  of  the  energy  of  a  molecule  is  no 
longer  tenable.  For  the  oscillators  assumed  by  Ein- 
stein we  have  the  evidence  that  equations  derived 
upon  that  assumption  indicate  variations  in  specific 
heat  in  the  direction  and  of  the  magnitude  confirmed 
by  experiment.  The  further  assumption  of  Einstein's 
equation,  that  energy  may  be  absorbed  only  in  quanta, 
was  originally  advanced  by  Planck  to  explain  certain 
phenomena  of  radiation.  The  success  of  the  applica- 
tion of  this  concept  of  quanta  to  the  phenomena  of 
specific  heat  is  at  once  a  confirmation  of  the  "  quantum 
theory"  and  a  promise  of  an  ultimate  explanation  of 
these  phenomena.  At  present,  however,  it  is  not 
possible  to  picture  with  definiteness  the  electronic 
construction  or  behavior  of  these  oscillators.  The 
quantum  theory  is  in  the  hands  of  the  future,  but  its 
fundamental  assumption  is  completely  verified :  energy 
may  only  be  absorbed  or  transferred  in  discrete  quanta. 


MOLECULAR  ENERGY  303 

Matter  and  energy,  the  two  realities  of  science,  are 
both  " granular." 

In  this  theory  we  have  the  modern  meeting  point 
of  physics,  chemistry,  and  physical  chemistry.  Origi- 
nally derived  to  explain  certain  difficulties  met  by 
older  theories  in  the  electromagnetic  radiation  of  light, 
it  has  been  applied  to  the  quantitative  expression  of 
specific  heats  of  solids  in  an  equation  involving  the 
physical  constants  met  with  in  the  study  of  gases. 
This  equation  may  therefore  be  applied  to  the  deter- 
mination of  the  number  of  molecules  per  mole.  The 
method  used  consists  first  in  determining  from  the 
relations  expressed  in  the  laws  of  thermodynamics  an 
expression  for  the  energy  of  radiation  of  a  "  black 
body"  in  terms  of  the  total  energy  of  the  body.  For 
the  total  energy  the  value  expressed  in  terms  of  elec- 
tronic oscillators  is  then  substituted.  The  actual 
radiation  is  measured  by  observing  its  heating  effect. 
The  value  of  N,  thus  determined,  is  quoted  as  64X1022 
in  the  table  of  page  282. 


CHAPTER  XXII 

ELECTRONIC  MAGNITUDES 

IN  Chapter  XIV  the  conduction  of  electricity  through 
highly  rarefied  gases  was  explained  in  terms  of  elec- 
trons. The  phosphorescence  of  the  sides  of  a  glass 
tube  in  which  such  conduction  is  taking  place  is  due 
to  the  impacts  of  electrons  proceeding  with  high 
velocities  away  from  the  cathode.1  Before  it  was 
proved  that  this  stream  was  corpuscular  it  was  assumed 
that  it  was  a  new  kind  of  "light,"  that  is,  an  electro- 
magnetic radiation,  and  hence  the  misnomer  "cathode 

rays."  For  experimental 
purposes  the  stream  may  be 
made  a  band  (indicated  by 
the  dotted  line  in  Fig.  37) 
by  arranging  the  anode  A 
as  a  hollow  cylinder.  A 
Fi  7  magnetic  field  at  right  an- 

gles to  this  ray  was  found 

to  bend  it  in  the  same  direction  as  we  would  now 
expect  with  our  knowledge  that  it  is  a  stream  of  elec- 
trons. By  deflecting  it  into  the  metal  vessel  V, 
which  was  connected  to  an  electroscope,  it  was  found 
that  the  ray  carried  a  negative  charge.  That  it  is  a 
stream  of  negative  particles  was  further  indicated  by 
the  fact  that  it  undergoes  a  deflection  in  the  proper 

1  Or  in  some  cases  to  ultra-violet  light  produced  at  the  cathode. 

304 


ELECTRONIC   MAGNITUDES  305 

direction  when  a  transverse  electrical  field  is  impressed 
upon  it. 

The  force  deflecting  the  stream,  when  it  is  subjected 
to  a  magnetic  field  of  intensity  H,  is  obtained  from  equa- 
tion (3)  of  page  207  as  F  =  qvH  if  q  represents  the  total 
quantity  of  electricity  transferred  along  a  length  L 
of  the  path  in  a  time  t  (and  hence  with  a  velocity  of 
v  =  L/t).  Similarly  if  the  electrical  field  intensity  is 
E  the  force  acting  on  the  quantity  q  which  is  con- 
tained by  this  length  of  beam  is  F  =  Eq.  The  velocity 
of  the  particles  was  measured  by  J.  J.  Thomson  by 
opposing  the  actions  of  these  two  fields  and  adjust- 
ing their  values  until  there  was  no  deflection,  hi  which 
case  Hvq  equals  Eq  and  v  is  E/H. 

Having  determined  hi  this  way  the  velocity  of  the 
electrons  in  a  particular  cathode  ray  it  was  possible 
to  find  the  ratio  of  the  charge,  e,  on  each  electron  to 
its  mass,  m,  by  the  deflection  of  the  ray  under  the  ac- 
tion of  an  electrical  field  only.  The  case  is  exactly 
analogous  to  that  of  a  bullet  shot  in  a  horizontal  line 
with  a  velocity  of  vt  except  that  the  medium  in  this 
case  is  ether  and  offers  no  friction.  If  the  accelera- 
tion at  right  angles  to  this  motion  is  a,  then  in  a  time 
t  the  bullet  will  travel  downward  the  distance  s  =  at2/2 
and  horizontally  the  distance  L  =  vt,  following  a  para- 
bolic path.  Now  a  is  always  F/m,  and  this  was  Ee/m 
in  the  experiment  and  hence  s  was 


~ 


Of  these  terms  s  and  L  are  directly  observable  and  v 
is  known  from  the  preceding  experiment;   hence  e/m 


306         TR-E   REALITIES  OF   MODERN   SCIENCE 


was  obtained.  The  numeric  representing  it  is  the 
number  of  units  of  quantity  per  gram.  It  was  found 
to  be  about  1.7X107  when  the  quantity  was  measured 
in  electromagnetic  units. 

Upon  the  assumption  that  the  ion  of  hydrogen  in 
electrolysis  is  essentially  the  mass  ra'  of  a  hydrogen 
atom  and  represents  an  equal  (but  opposite)  charge 
of  electricity,  the  value  of  m  was  obtained.  In  elec- 
trolysis one  e.m.  unit  of  quantity  is  transferred  by 
(1.008)(0.01118)/(107.88)  or  about  10~4  gram  of  hydro- 
gen. Hence  e/m'  is  104  and  a  value  of  m  as  approxi- 
mately 1/1700  part  of  m'  was  thus  obtained. 

Methods  for  determining  the  charge  e  were  soon 
devised.  One  of  these,  by  Townsend,  proved  to  be 
basic  to  most  of  the  subsequent  determinations.  In 
Chapter  XVIII  we  spoke  as  though  all  the  molecules 
of  gas  which  rise  from  the  electrodes  of  an  electrolyte 
were  neutral.  As  a  matter  of  fact,  perhaps,  one  in  a 
million  may  carry  a  charge.  The  sign  of  the  latter 
depends  upon  the  electrolyte  from  which  the  gas  rises. 
Now  these  charged  molecules,  when  in  air  containing 
water  vapor,  become  nuclei  of  small  drops,  aggregating 
to  themselves  water  molecules  and  forming  a  visible 
cloud.  The  natural  assumption  is  that  in  such  a 
condensation  the  number  of  droplets  is  equal  to  the 
number  of  unneutral  gas  molecules.  The  charge  per 
unit  volume  Townsend  found  by  allowing  the  gas  to 
give  up  its  charge  to  an  electrometer  (a  calibrated 
electroscope).  The  average  radius  of  the  droplets 
he  found  by  observing  the  time  it  took  the  cloud  to 
settle,  under  the  action  of  gravity,  through  a  known 
distance  and  then  applying  Stokes's  Law.  The  weight 


ELECTRONIC   MAGNITUDES 


307 


of  the  cloud  per  unit  volume  was  obtained  by  passing 
it  through  drying  tubes  and  observing  their  increase 
in  weight.  From  this  and  the  size  of  the  particles  he 
found  the  number,  and  upon  the  assumption  that 
each  is  abnormal  by  only  one  electron  he  obtained  from 
his  electrometer  reading  the  value  of  e. 

The  most  exact  determination  of  the  electronic 
quantity  is  that  of  Millikan.1  The  method  which  he 
devised  involved  the  observation  of  the  motion  of  a 
small  particle  of  oil  (about  10"4  cm.  diam.)  under  the 
action  of  gravity  and  of 
an  electrical  field.  The 
particle  usually  carried 
an  electrical  charge  pro- 
duced "by  friction"  hi 
its  own  formation,  for 
the  drops  were  obtained 
by  blowing  carefully  fil- 


jj  "? 

Larth  Earth 


tered    air    through    an  Earth 

ordinary  atomizer  con- 
taining the  oil.  The  electrical  field  was  maintained 
between  two  parallel  plates,  M  and  N  of  Fig.  38,  by 
means  of  the  battery  B.  The  switch  St  if  to  the  right, 
permitted  the  plates  to  be  raised  to  the  difference  of 
potential  of  the  battery  or  to  discharge  through  the 
connecting  wires  in  case  it  was  moved  to  the  left.  The 
four  circles  at  the  right  of  the  figure  represent  a  com- 
mutator, a  device  of  four  mercury  cups,  two  of  which 
connect  to  the  battery  and  the  other  two  to  the  parallel 
plate  condenser  (in  one  case  through  the  earth) .  These 
mercury  cups  could  be  connected  hi  any  desired  way 
»Cf.  Millikan,  "The  Electron,"  Univ.  of  Chicago  Press,  1917. 


308         THE   REALITIES  OF   MODERN  SCIENCE 

by  short  wires  and  thus  the  polarity  of  the  condenser 
could  be  reversed. 

Above  the  plates  was  a  closed  chamber,  which  re- 
ceived the  spray  from  the  atomizer ;  and  occasionally 
a  small  drop  would  find  its  way  through  the  pinhole, 
p,  in  the  center  of  the  upper  plate  and  thus  be  avail- 
able for  observation.  Powerful  beams  of  light  were 
directed  through  the  space  between  the  plates  from 
the  windows  c  and  c  on  the  right  and  left.  The  illu- 
minated drop  was  observed  by  a  small  telescope  through 
a  window  in  the  apparatus.  In  the  eyepiece  of  the 
telescope  were  small  parallel  hairs  which  served  as 
fixed  reference  lines  for  noting  the  position  of  the 
"  droplet."  This  droplet  appeared  merely  as  a  speck  of 
light,  like  a  bright  star  on  a  black  background. 

If  it  was  desired  to  alter  the  charge  on  an  oil  drop 
the  air  between  the  plates  was  ionized  by  exposure  to 
some  ionizing  agent,  either  X-rays  or  radium.  If  it 
was  desired  to  give  a  drop  an  additional  negative  charge 
it  was  arranged  that  it  should  be  near  the  positive 
plate  of  the  condenser  while  the  air  was  being  ionized. 
Toward  this  plate  there  came,  of  course,  a  shower  of 
electrons  produced  from  the  air  by  the  ionization. 
Conversely,  when  it  was  desired  to  give  the  droplet 
an  additional  positive  charge  it  was  held  near  the  nega- 
tive plate,  and  thus  it  suffered  collisions  from  positive 
ions. 

The  position  of  the  drop  could  be  controlled  at  will 
because  it  was  so  small  that  it  fell  very  slowly  indeed. 
For  example,  a  matter  of  ten  or  twelve  seconds  would 
be  required  for  the  droplet  to  fall  under  the  action  of 
gravity  through  the  distance  of  about  one  centimeter 


ELECTRONIC   MAGNITUDES  309 

between  the  plates.  On  the  other  hand,  if  it  carried 
a  negative  charge,  it  could  be  made  to  move  up  by 
making  the  upper  plate  positive  with  respect  to  the 
lower  (by  the  proper  motion  of  the  switch  and  the 
commutator).  Similarly,  if  the  droplet  were  positive, 
the  upward  motion  would  be  accomplished  by  reversing 
the  battery  connection. 

We  must  remember  that  the  little  drops  with  which 
Millikan  worked  were  so  small  that  they  would  only 
fall  about  a  millimeter  a  second  and  yet  they  were  a 
thousand  times  or  more  the  diameter  of  the  molecules 
of  the  ah*  through  which  they  fell.  The  change  in 
the  inertia  of  the  drop  due  to  its  picking  up  a  molecule 
of  the  air  in  a  collision  would  then  be  negligible.  Now 
Millikan  observed  thousands  of  these  drops  at  various 
times  and  would  sometimes  observe  the  actions  of 
a  single  drop  for  hours.  By  allowing  it  to  fall  and 
then  by  applying  the  potential  and  moving  it  up  again 
he  could  keep  it  in  the  field  of  his  telescope.  When  he 
wanted  it  to  fall  he  short-circuited  the  plates,  thereby 
destroying  the  electrical  field  between  them.  He  made 
observations  of  the  tune  of  fall  and  the  time  of  rise 
consecutively. 

The  time  of  fall  for  a  given  droplet,  when  there  was 
no  electrical  field,  was  found  to  be  always  the  same, 
within  the  limits  of  the  experimental  error  involved 
in  observing  the  times  when  it  passed  the  cross  hairs 
of  the  telescope.  The  effect  of  the  charges  which  the 
drop  carried  hi  increasing  the  resistance  of  the  gaseous 
medium  was  therefore  inappreciable.  There  was  no 
appreciable  acceleration  under  the  action  of  gravity, 
and  the  motion  of  the  drop  was  that  of  a  body  hi  a 


310         THE   REALITIES   OF   MODERN   SCIENCE 

viscous  fluid,  where  the  velocity  is  proportional  to  the 
force.  In  the  case  of  the  fall,  the  velocity,  v\,  was 
therefore  directly  proportional  to  mg,  the  force  of 
gravitation. 

The  time  of  rise  would  be  found  to  be  the  same  for 
several  trips  and  then  suddenly  it  would  change,  in- 
dicating a  change  in  the  charge  carried  by  the  droplet. 
The  velocity  v2  of  the  motion  upward  is  also  propor- 
tional to  the  force.  This  force  is  Fne—mg,  where  F 
is  the  force  due  to  the  field  maintained  by  the  battery 
upon  one  electrostatic  unit  of  quantity,  e  is  the  charge 
in  e.s.  units  of  an  electron,  and  n  is  the  number  of  elec- 
trons carried  by  the  droplet.  If  the  number  of  electrons 
is  altered  to  nf,  then  the  velocity  v2  will  be  altered,  say 
to  the  value  v2'.  Thus 

(1) 

— mg  (2) 

but 

Vt'&Fn'e  —  mg  (3) 

Hence 

(ri  -n)e  =  (v2'-  v2}mg/Fvl  (4) 

That  is,  the  change  in  the  number  of  electrons  carried 
by  the  droplet  should  be  proportional  to  the  change 
in  the  velocity  which  is  thereby  occasioned,  provided 
that  the  value  of  F  is  maintained  constant. 

The  changes  in  velocity  which  Millikan  observed 
always  bore  simple  ratios  to  each  other.  He  found, 
in  other  words,  that  there  was  one  minimum  value  for 
(v2  —  v2)  and  that  all  the  other  values  were  simple  multi- 
ples. That  could  only  mean  that  the  change  in  charge 
was  due  to  the  additions  or  subtractions  of  a  definite 


ELECTRONIC   MAGNITUDES  311 

amount  of  electricity  or  of  some  whole  number  of  times 
this  amount.  His  observations  are,  therefore,  to  be 
taken  as  a  final  demonstration  of  the  existence  of  elec- 
tricity only  in  definite  amounts.  This  definite  amount 
of  electricity  he  found  to  be  independent  of  the  manner 
in  which  it  was  produced,  as  for  example  by  friction, 
or  by  the  ionization  of  gases.  It  is  independent  of 
the  substance  from  which  it  is  obtained  or  with  which 
it  is  later  associated,  since  he  used  for  his  drops  con- 
ductors like  mercury,  poor  conductors  like  glycerin, 
and  non-conducting  oil. 

If  for  (nr — ri)  there  is  substituted  unity,  and  for 
(%'  —  #2)  the  smallest  change  in  velocity  due  to  a  change 
in  charge,  the  value  of  e  may  be  computed  from  equa- 
tion (4)  above,  providing  m  is  known.  Stokes's  Law 
for  a  falling  drop,  as  given  in  equation  (2),  page  275,  may 
be  solved  to  give  the  radius  of  the  drop  in  terms  of  the 
velocity  and  viscosity,  as  follows : 


r  = 


2(D-d)g 


(5) 


Now  the  numerator  of  the  expression  given  in  Stokes's 
equation  is  mg,  since  4r3(D  -d)g/3  is  the  weight  of 
the  sphere  in  a  vacuum  less  the  weight  of  the  liquid 
displaced.  Hence,  making  this  substitution  in  equa- 
tion (4)  and  also  substituting  r  from  equation  (5),  gives 

_4/9c\f/      TT 

•sir)  (rtp^ 

Millikan,  however,  found  that  the  values  he  obtained 
for  e  by  using  this  equation  depended  upon  the  size 
of  the  drop,  being  larger  as  the  drop  experimented  with 


312         THE   REALITIES  OF   MODERN  SCIENCE 

was  smaller.  He  therefore  doubted  the  applicability 
of  Stokes's  Law  to  spheres  of  such  a  small  size.  One 
of  his  students,  Arnold,  tested  Stokes's  Law  for  a  wide 
range  of  diameters  by  using  small  spheres  formed  of 
rose-metal,  a  substance  with  a  low  melting  point. 
The  velocity  was  found  to  be  correctly  expressed  by 
Stokes's  Law  only  when  the  radius  r  of  the  sphere  was 
large  as  compared  to  the  mean  free  path  of  the  mole- 
cules of  the  air.  The  mean  free  path  being  inversely 
as  the  pressure  of  the  air  the  true  velocity  was  found 
to  be  expressed  as 


pr, 

where  p  is  the  pressure,  A  is  a  constant,  and  Vi  is 
the  velocity  of  fall  as  calculated  by  Stokes's  Law. 

Millikan  therefore  determined  the  value  of  e  by 
using  droplets  in  air  under  various  pressures,  and  thus 
found  not  only  the  value  of  the  correction  factor  A 
but  also  the  true  value  of  e.  As  a  further  check  he 
repeated  his  experiments  with  drops  so  large  that 
Stokes's  Law  would  hold  for  fall  through  air  at  atmos- 
pheric pressure.  He  then  obtained  e  by  substituting 
directly  in  equation  (4). 

The  value  of  e  as  obtained  by  this  method  is 
4.774X10-10  electrostatic  unit  or  1.591X10-20  electro- 
magnetic unit.  This  determines  Avogadro's  constant, 
N,  as  6.065X10  3.  From  this  the  mass  of  a  hydrogen 
atom  is  obtained  as  1.662X10"24  gram.  Using  the 
best  value  of  e/m  for  the  electron  and  the  above  value 
of  e  gives  the  mass  of  the  electron  as  1/1845  part  of  the 
hydrogen  atom,  or  9.01  X10~28  gram.  Upon  the  as- 
sumption that  the  entire  mass  is  electromagnetic  and 


ELECTRONIC   MAGNITUDES  313 

that  this  charge  is  uniformly  distributed  over  a  sphere, 
the  radius  of  the  electron  is  indicated  as  about  2X10~13 
cm.  Such  magnitudes  are  quite  beyond  our  compre- 
hension except  by  analogies,  such  as  a  statement  of 
how  long  it  would  take  if  every  man  in  the  world  should 
count  before  the  total  number  counted  equaled  the 
number  of  electrons  equivalent  in  inertia  to  one  gram. 
They  may  be  more  nearly  visualized  after  consider- 
ing some  phenomena  of  ionization  of  gases. 

For  experiments  involving  ionization  X-rays  may  be 
used  (cf.  page  190)  or  the  gas  may  be  exposed  to  a 
radioactive  substance.  The  radiations  from  the  latter 
are  of  three  types,  although  all  three  do  not  usually 
occur  in  the  case  of  a  single  substance.  They  are 
usually  designated  by  alpha,  beta,  and  gamma  re- 
spectively. In  this  case  also  the  terms  "  radiation " 
and  "rays"  are  misnomers,  at  least  as  far  as  concerns 
the  first  two,  for  the  a  rays  are  formed  of  a  stream  of 
positively  charged  helium  atoms  and  the  ft  rays  by 
electrons.  The  7  rays  are  really  a  radiation,  being  a 
penetrating  radiation  of  the  nature  of  a  pulse  like  the 
X-rays. 

The  disintegration  of  an  atom  of  radium,  for  example, 
results  in  the  projection  in  opposite  directions  of  a 
helium  atom  (deficient  by  one  electron)  and  a  niton 
atom,  with  velocities  inversely  as  the  masses.  The 
lighter  a  particles  have  about  100  times  the  greater 
velocity,  and  travel  about  seven  tenths  as  fast  as  does 
light.  The  niton  atoms,  themselves,  disintegrate 
further  and  after  successive  changes  in  which  further 
a  particles  are  lost,  as  well  as  electrons,  a  more  stable 
resultant  of  polonium  is  reached. 


314         THE   REALITIES   OF   MODERN  SCIENCE 

All  three  types  of  rays  produce  ionization.  The 
rays  themselves  may  be  separated  by  subjecting  them 
to  a  transverse  electrical  field,  for  they  consist  of 
streams  of  positive  and  negative  corpuscles,  which 
will  be  oppositely  deflected,  and  a  true  radiation  which 
suffers  no  deviation.  Their  ionizing  effects  are  easily 
traced  in  a  gas  containing  water  vapor  by  the  phenome- 
non of  condensation  about  the  newly  formed  ions. 
The  paths  may  therefore  be  photographed.  Fig.  39 
illustrates  the  case  of  a  particles  projected  from  radium 
through  air  and  Fig.  40  is  an  enlargement  of  a  portion 
of  the  preceding  figure.  Fig.  41  shows  the  path  of  a 
13  particle. 

The  paths  shown  by  the  a  particles  are  from  3  to  7 
cm.  long.  Now  in  each  cubic  centimeter  of  air  there 
are  about  2.7  X1019  molecules,  so  that  each  of  the 
particles  traced  by  the  photograph  must  have  ionized 
millions  of  air  molecules.  Through  the  electronic 
systems  which  form  these  molecules  each  a  particle 
passed  so  rapidly  as  not  to  be  deflected  although  it 
repeatedly  knocked  off  electrons.  In  two  cases  in  Fig.  40 
there  may  be  seen  sharp  changes  in  the  direction, 
when  one  probably  collided  head-on  with  the  massive 
nucleus  of  an  air  molecule.  These  paths  would  in- 
dicate that  the  nucleus  of  a  molecule  is  relatively 
very  small  indeed,  for  otherwise,  in  all  probability, 
there  would  have  been  many  more  such  collisions. 

In  Fig.  41,  from  bottom  to  top,  we  trace  the  ionization 
of  a  high  speed  electron  (/3  particle).  It  does  not 
ionize  as  frequently  but  it  also  follows  a  straight  line. 
Its  velocity  is  obviously  so  high  that  it  does  not  re- 
main long  enough  within  the  neighborhood  of  the 


FIG.  39. 


FIG.  40. 


ELECTRONIC   MAGNITUDES  315 

individual  systems  which  it  passes  to  receive  by  inter- 
action a  component  of  velocity.  On  the  other  hand, 
in  Fig.  42  are  given  the  paths  of  some  electrons  with 
smaller  velocities  which  ionize  more  frequently  but 
are  also  deflected.  The  slower  speed  electron  ionizes 
a  larger  number  of  molecules  per  centimeter  than  does 
the  higher  speed  particle.1  Both  are  gradually  retarded, 
and  below  a  certain  speed  ionization  does  not  occur. 

In  the  cases  of  Fig.  40  and  Fig.  41  the  electrons 
which  are  knocked  off  from  the  molecules  leave  at 
such  small  speeds  that  they  may  not  themselves  serve 
for  ionizing  agents  (unless  accelerated  by  an  impressed 
electrical  field  as  was  discussed  in  Chapter  XIV). 
In  Fig.  42,  however,  is  shown  the  trace  of  an  X-ray. 
This  radiation  does  not  merely  jar  loose  electrons  as 
did  the  a  and  ft  particles,  but  seems  to  shake  them 
loose  with  such  violence  that  each  electron  leaves 
behind  itself  a  trail  of  ionization. 

The  ability  of  electromagnetic  radiations  to  shake 
out  electrons  is  not  limited  to  X-rays  and  7  rays  but, 
as  we  saw  in  Chapter  XIV,  is  characteristic  also  of 
ultra-violet  radiations.  Metals,  especially  zinc  and 
the  alkali  metals,  emit  electrons  when  exposed  to 
ultra-violet  light.  The  effect,  known  as  photoelectric, 
was  recognized,  before  the  theory  of  electrons  was  so 
fully  developed,  by  the  fact  that  under  these  condi- 
tions the  metal  acquires  a  positive  charge. 

In  this  phenomenon  are  some  curious  facts  which 
have  supported  Planck's  suggestion  that  energy  (in  this 

1  Millikan  estimates  that  this  particle  (Fig.  41)  on  the  average 
passed  through  10,000  gas  molecules  for  every  time  it  reacted  suf- 
ficiently to  jar  loose  an  electron.  Cf.  Millikan,  "The  Electron,  "p.  186. 


316         THE   REALITIES  OF   MODERN  SCIENCE 

case  radiant)  is  absorbed  only  in  discrete  quanta. 
It  is  found  that  if  the  frequency  of  the  light  is  main- 
tained constant  the  number  of  electrons  emitted  in- 
creases with  the  intensity  of  the  illumination,  but 
that  the  speed,  and  hence  the  energy,  of  each  electron 
is  independent  of  this  intensity.  In  other  words  more 
quanta  are  received  if  the  light  is  more  intense,  and 
hence  more  electrons  can  be  emitted,  but  each  can 
only  absorb  the  same  quantum  as  before  and  hence 
must  possess  only  that  amount  of  energy.  On  the 
other  hand,  if  the  intensity  of  the  light  is  kept  con- 
stant and  the  frequency  is  increased,  the  quanta  are 
greater,  as  is  manifested  by  the  increased  velocity  of 
the  emitted  electrons. 

The  photoelectric  effect,  therefore,  offers  a  very 
convenient  method  of  determining  Planck's  constant, 
h,  by  which  the  frequency  must  be  multiplied  to  ob- 
tain the  quantum1  of  energy  corresponding  to  that 
frequency.  The  determination  involves  a  knowledge 
of  the  charge  on  the  electron.  This  constant  was 
therefore  redetermined  by  Millikan  using  his  value  for 
e.  He  found  it  to  be  6.56  X  10~27. 

1  A  frequency  of  about  1.5  XlO14  is  that  at  which  a  heated  body 
radiates  the  maximum  energy.  This  corresponds  to  a  wave  length 
of  (3.0X1010)/1.5X1014,  or  about  two  thousandths  of  a  milli- 
meter, or  2/i  as  it  is  abbreviated.  The  quantum  is  then  (1.5xl014) 
(6.56X10-27)  or  9.9X10-13  erg.  The  lowest  frequency  visible  as 
red  light  is  3.75X1014,  of  wavelength  X=0.8/x,  and  its  quantum  is 
2.5xlO~12  erg.  The  highest  frequency  visible  as  violet  light  is 
7.5xl014,  that  is  \=OAn,  and  the  quantum  is  5.0xlO~12erg.  Ultra- 
violet light  is  not  transmitted  by  ordinary  glass  as  well  as  by 
quartz.  About  the  highest  frequency  the  latter  transmits  is 
1.5xl015,  that  is  X=0.2/x,  and  the  corresponding  quantum  is  9.9X10"12 
erg. 


FIG.  41. 


FIG-  42. 


ELECTRONIC   MAGNITUDES  317 

What  might  be  called  the  classical  theories  of  radia- 
tion have  been  upset  by  the  remarkable  accumulation 
of  evidence  indicating  the  correctness  of  Planck's 
hypothesis  that  energy  is  absorbed  only  in  definite 
quanta,  each  of  value  hf  where  /  is  the  frequency.  The 
fact  that  it  is  so  absorbed  is  evident  from  the  photo- 
electric effects,  but  the  mode  of  its  emission,  transmis- 
sion, and  absorption  is  as  yet  in  the  domain  of  specula- 
tive physics.  For  that  reason  we  shall  not  attempt  a 
further  discussion  of  radiation.  Future  scientific  in- 
vestigations and  analyses  will  ultimately  reconcile  the 
quantum  theory  with  the  portion  of  the  classical 
theory  which  is  supported  by  direct  experiment.  One 
of  the  difficult  points  is  to  reconcile  the  idea  of  inter- 
ference of  wave  trains  of  continuous  energy  with  the 
present  evidence  of  the  granular  nature  of  the  energy. 
The  phenomenon  of  interference  was  responsible  for 
overturning  that  earlier  corpuscular  theory  of  light 
which  had  the  influential  support  of  Newton.  That 
theory  pictured  light  as  a  stream  of  corpuscles,  while 
the  present  theory  may  only  definitely  assert  that  the 
energy  which  we  recognize  as  light  is  to  be  received 
by  an  electron  oscillator  only  in  definite  quanta. 

With  one  application  of  the  phenomenon  of  inter- 
ference we  are  immediately  interested  because  it  is 
involved  in  the  determination  of  the  electron  numbers 
which  are  referred  to  in  Chapter  VIII.  Interference, 
however,  is  a  phenomenon  common  to  all  instances  of 
the  transmission  of  energy  from  periodic  sources.  It 
occurs  in  molecular  media  as  well  as  in  the  ether. 
Consider  for  example  a  row  of  similar  sources  as 
indicated  in  Fig.  43.  Along  the  line  ad  the  magni- 


318         THE   REALITIES  OF   MODERN  SCIENCE 

tude  of  the  effect  of  these  will  differ  from  point  to 
point.  Thus  suppose  that  at  a  the  disturbance  is  a 
maximum.  At  some  point  like  fr,  however,  it  will 
happen  that  the  disturbance  received  from  any  source, 
as  sit  will  be  Just  opposite  in  its  effect  to  that  received 
from  the  adjacent  source,  s2.  Similarly  for  other 


pairs  of  sources.  At  b  there  is  therefore  complete  inter- 
ference. At  another  point,  as  c,  the  effects  will  again 
be  in  phase  and  there  will  be  reenforcement  and  a 
consequent  maximum.  The  distance  ac  is  dependent 
upon  the  frequency  of  the  several  identical  oscillators. 
If  these  individually  give  rise  to  more  than  one  fre- 
quency there  will  be  a  different  series  of  points  like 
b  and  c  for  each  frequency,  with  the  result  that  the 
location  of  the  maximum  will  no  longer  be  a  point  but 
will  appear  as  a  spectral  band.1 

1  The  phenomenon  is  easily  observed  if  one  makes  in  a  visiting 
card  two  small  pinholes  separated  by  about  half  the  diameter  of  a 
pinhead.  Holding  this  card  close  to  the  eye  and  observing  a  dis- 
tant and  intense  source  of  light,  e.g.  a  street  lamp,  the  two  holes 
act  like  two  sources  in  the  diagram  of  Fig.  43  and  the  line  ad  is 
the  retina.  A  series  of  spectral  bands  should  then  be  observed. 
If  it  is  not,  the  pinholes  are  probably  too  large  or  too  far  apart,  or 
both.  The  pattern  seen  through  the  meshes  of  an  umbrella  while 
looking  at  a  distant  light  is  a  familiar  illustration. 


ELECTRONIC   MAGNITUDES  319 

The  sources  of  Fig.  43  may  be  secondary  sources, 
all  receiving  energy  from  some  distant  source.  If  they 
are  excited  by  monochromatic  light  its  frequency  may 
be  determined,  provided  the  separation  of  the  sources 
Si,  s2,  the  distance  Si  a  and  the  separation  of  the  bands 
are  measurable.  This  principle  has  been  applied  to 
the  measurement  of  the  frequency  of  visible  light  by 
using  the  so-called  diffraction  grating,  a  smooth  surface 
of  speculum  metal  upon  which  are  ruled  equidistant 
scratches.  When  this  is  illuminated  by  a  beam  of 
light,  the  intervening  spaces  act  by  reflection  as  sources. 
These  sources  are  lines  and  not  points  and  result  in 
spectral  bands  parallel  to  themselves,  as  may  be  seen 
by  considering  the  sources  of  Fig.  43  to  be  points  in  a 
cross  section  of  such  a  grating.  The  grating  thus 
serves  to  resolve  light  which  is  not  monochromatic  into 
its  monochromatic  components,  since  the  location  of 
each  band  will  depend  upon  its  frequency. 

Conversely,  if  the  primary  source  is  monochromatic, 
an  indication  of  the  spacing  of  the  point  sources  is 
obtained  from  the  separation  of  the  bands.  If  these 
point  sources  do  not  lie  in  the  same  line,  as  we  have 
so  far  assumed,  but  are  in  three  dimensions,  the  result- 
ing interference  bands  will  form  a  complicated  pattern 
from  which,  however,  some  indication  of  the  configura- 
tion of  the  sources  may  be  obtained.  Of  course,  the 
spacings  of  the  sources  must  be  regular,  and  not  hap- 
hazard, if  this  is  to  be  done.  Such  a  regular  structure 
is  afforded  by  the  atoms  of  crystalline  substances,  but 
their  separations  are  very  small  and  a  diffraction  pat- 
tern is  impossible,  unless  the  wave  length  of  the  light 
which  is  used  is  smaller  than  that  of  visible  light.  In 


320         THE   REALITIES  OF   MODERN  SCIENCE 

X-rays,  however,  sources  of  short  wave  length  are 
available,  and  for  such  radiations  crystals  act  like 
diffraction  gratings,  the  interference  patterns  of  which 
may  be  made  visible  by  photographing. 

If  various  crystals  are  thus  examined  under  excitation 
by  a  given  source  of  X-rays,  very  definite  indications 
of  their  molecular  structure  are  obtainable.1  Con- 
versely, if  a  given  crystal  is  used  as  a  grating  and  is 
illuminated  by  X-rays  from  different  sources,  indica- 
tions may  be  obtained  as  to  the  frequency  of  the  sources. 

The  principle  was  employed  by  Moseley  to  deter- 
mine the  characteristic  radiations  of  various  sub- 
stances when  used  as  the  anti-cathode  of  an  X-ray 
tube.2  In  other  words,  the  sources  were  the  electronic 
oscillators  of  the  substance.  These  were  excited  by 
a  bombardment  of  electrons  from  the  cathode  of  a 
highly  evacuated  tube.  The  spectral  bands  obtained 
in  this  way  are  illustrated  in  Fig.  44.  As  the  substance 
of  the  anti-cathode  is  changed  from  arsenic  (As)  to 
selenium  (Se)  or  from  rubidium  (Rb)  to  strontium  (Sr) 
there  occurs  the  same  shift  of  the  characteristic  spec- 
trum shown  in  the  figure.  This  indicates  that  the  group 
of  oscillators,  constituting  the  atom,  changes,  as  the 
substance  is  changed  to  that  next  in  the  series  of 
elements,  by  the  addition  of  the  same  number  of  oscil- 
lators, that  is  by  the  same  charge.  The  number  of 
positive  electrons  in  the  nucleus  of  the  atom  of  any 
element  is  thus  to  be  determined  from  the  number  in 
that  of  any  other  element  by  counting  the  number  of 

*Cf.  W.  H.  Bragg  and  W.  L.  Bragg,  "  X-rays  and  Crystal  Struc- 
ture," G.  Bell  &  Sons,  Ltd.,  London,  1915. 

2  Cf.  Kaye,  "X-Rays."     Longmans,  Green  and  Co.,  1914. 


Sf 

}7 

II 

41 

45 


fib 


FIG.  44. 


ELECTRONIC  MAGNITUDES  321 

steps  required  to  displace  the  spectrum  of  the  second 
to  coincide  with  that  of  the  first.  The  characteristic 
lines  of  the  hydrogen  atom  appear  displaced  in  the 
spectra  of  the  other  elements.  (This  is  evidence  in 
favor  of  the  concept  of  a  positive  electron  which  is  the 
nucleus  of  the  hydrogen  atom.)  Starting  with  hydro- 
gen as  unity  the  atomic  numbers  of  a  large  number  of 
elements  have  been  obtained  by  the  analysis  of  these 
and  similar  experiments.  The  method  offers  a  pre- 
cision in  excess  of  that  of  preceding  methods  and  has 
definitely  established  these  numbers. 

Earlier  experiments  indicating  atomic  numbers 
made  use  of  the  deflection  of  an  a-ray  due  to  collisions 
(cf .  page  314) .  Calculations  may  be  made  of  the  charge 
of  the  nucleus  of  a  molecule  from  the  scattering  of  a 
particles  (or  of  ft  particles)  in  their  passage  through 
solids.  Using  thin  films  of  metals  Rutherford  l  de- 
termined some  atomic  numbers  which  are  hi  general 
agreement  with  more  recent  values. 

The  concept  of  an  electronic  constitution  of  matter, 
which  we  have  sketched  in  the  preceding  chapters,  is 
to-day  commonly  accepted.  Its  development  has  been 
so  rapid  that  few  scientists  besides  those  actively  en- 
gaged in  its  research  have  as  yet  adopted  its  terminol- 
ogy and  the  necessary  point  of  view.  In  terms,  however, 
of  electrons  and  quanta,  the  scientific  progress  of  the 
next  few  years  will  undoubtedly  be  expressed. 

*Cf.  Rutherford,  "Radioactive  Substances  and  Their  Radia- 
tions."   Cambridge  University  Press,   1913. 


INDEX 


absolute  motion,  64 

absolute  zero,  167,  171 

acceleration,  134,  144 

acid,  255 

actinium,  92 

action  and  reaction,  148 

active  mass,  260 

Ahmes  papyrus,  120 

alchemy,  73,  85,  152 

Alexandria,  50 

alpha  rays,  313 

ampere,   178 

amplitude,  33 

angle  of  repose,  17 

anion,  245 

ankylosis,  295 

anode,   189 

anti-cathode,  190 

Arabs,  50,  73 

Archimedes'  principle,  41 

Aristotle,  34,  40,  52 

Arnold,  312 

Assyrians,  27 

atomic  numbers,  91,  92,  320 

atomic  size,  90 

atomic  weight,  84,  91,  99 

atomists,  85 

atoms,  71,  72,  78,  80 

attraction  of  currents,  198,  201, 

availability  of  energy,  105,  153 

Avogadro,  80,  83,  84 

Avogadro's  constant,  312 

Babylonians,  10 

Bacon,  Roger,  41 

balance,  28 

base,  256 

Berzelius,  78,  84 

beta  rays,  313 

Black,  74,  76 

black  body  radiation,  299 

boiling  point,  171,  224,  239 

Boyle,  54,  74,  77,  78 


Boyle's  law,  57 

Bragg,  319 

Braun,  263 

brickbuilders,  11 

bronze  lion,  27 

Brown,  Robert,  158 

Brownian  movement,  158,  275 

by-products  of  experimentation,  53 

Caesar,  48 
calculus,  32 
calibration,  31 
calorie,  289 
calx,  75 

Cannizzaro,  80,  84 
Carnot  cycle,  269 
carriers  of  electricity,  184 
cathode,  189 
cathode  rays,  304 
cation,  245 
Cavendish,  74 
celestial  sphere,  34 
Celsius,  168 
center  of  mass,  148 
centi-,  26 

Centigrade  scale,  113,  168 
certificates  of  indebtedness,  183 
Charlemagne,  51 
207    Chinese,  52 

circular  motion,  195 

Clausius,  79,  269,  281 

clepsydra,  34 

clocks,  33 

Columbus,  52 

compass,  52 

concentration  of  solutions,  236,  260 

condensation,  pulse  of,  62 

conduction.     See  heat  or  electricity. 

conservation  of  energy,  110,  113,  141 

continuity,  molecular,  217 

correspondence  of  molecular   states, 

217 

Coulomb,  174,  178 
323 


324 


INDEX 


Coulomb's  law,  174 

critical  pressure,  233 

critical  temperature,  221,  232 

critical  volume,  232 

crystal  structure,  319 

Curie,  92 

current,  electrical,  192 

Dalton,  72,  74,  78,  79,  81,  82 

Dalton's  laws,  79,  234 

Davy,  73 

day,  solar,  34 

days  of  the  week,  11 

deductive  reasoning,  47 

degrees  of  freedom,  165 

Democritus,  70 

derived  units,  119,  121 

diameter  of  molecule,  281 

diffraction  grating,   319 

dissociation,  electrolytic,  248 

dog,  attitude  toward  fire,  1 

Doppler's  principle,  65 

double  weighing,  28 

Dulong    and    Petit's  law,   296,   302 

dyne,  145 

e,  307 

e/m,  305 
efficiency,  21 
Einstein,  298 
elasticity,  30 
elastic  limit,  31 
electricity,  88,  94,  174 
electricity,  conduction  of,  182,  184 
electricity,  positive  and  negative,  93 
electricity,  units  of,  177,  178,  206,  277 
electrification,  95,  173 
electrochemical  equivalent,  254 
electrolytes,  183,  244,  253 
electromagnetic  mass,  212 
electromagnetic  units,  277 
electromagnetic  waves,  124 
electromotive  force,  208 
electron,  88,  90,  97,  304 
electronic  oscillators,  297,  302 
electrons,  interactions  of,  195 
electrons,  motions  of,  173,  179 
electron,  theory,  34 
elements,  chemical,  72,  91 
e.m.f.,  208 
energy,  8,  101,  114 
energy  in  electrical  currents,  199,  209 


energy,  molecular,  161,  303 

engineer,  126 

England,  27 

entropy,  268 

equilibrium,  227,  229,  257,  259,  261 

equipartition    of    energy,    165,    271, 

294,  296 

equipotential   surface,    107 
erg,  144 

ether,  90,  191,  200 
Euclid,  44 
evaporation  of  solutions,  248 

faculty  of  accommodation,  266 

Fahrenheit,  168 

Fahrenheit  scale,  113 

falling  bodies,  46,  135 

Faraday,  208,  254 

Faraday's  laws  of  electrolysis,  254 

feudalism,  22 

fire,  2,  3 

fire  machine,  6 

fluxions,  32,  124 

foot,  25 

force,  139 

forced  vibration,  298 

Franklin,  93,  94 

freezing  point,  171,  239,  242 

French  Revolution,  122 

frequency,  61 

friction,  17,  113 

full  radiator,  299 

fundamental  units,  123 

"g,"  135,  147,  197 

Gaede  molecular  pump,  283 

Galileo,  32,  46,  54,  110,  135,  136,  166 

Galvani,  174 

gamma  rays,  313 

gas  constant,  R,  218 

gas  equation,  163 

gas  pressures,  low,  284,  287 

Gay-Lussac,  83 

Geissler  tube,  189 

Gilbert,  54,  57 

Gizeh,  22 

gram-centimeter,  108,  145  ' 

gram-molecule,  85 

gravitational  units,  143 

Greeks,  11,  85 

Gresham's  law,  21 

Guldberg,  259 

gunpowder,  52 


INDEX 


325 


heat  capacity,  166 

heat  conduction,  161,  185 

heat  dissipation,  electrical,  193 

heat,  specific,  288,  290,  292 

Hebrews,  22 

helium,  97 

Hertz,  124 

Hiero,  42,  50 

Hippocrates,  85 

Hooke,  54 

Huyghens,  35 

hydraulic  press,  59 

ideal  problems,  18 
inclined  plane,  17 
India,  27 
indicators,  261 
induced  currents,  208 
inductive  reasoning,  47 
inertia,  101 
infinitesimals,  130 
infra  red,  187 

intra-molecular  motions,  162 
inventor,  126 
ionization,  180,  181,  314 
ionization  constant,  259 
ionogen,  244 
isothermal,  218,  227 

J,  289 

Joule,  70,  112,  289 

joule,  145 

Joule's  laws,  193,  220 

Karlsruhe,  80 

kilo-,  26 

kilogram,  27,  123 

kinetic  energy,  102 

kinetic  energy,  electronic,  214 

kinetic  energy,  molecular,  153,  217 

Langmuir,  286 

latent  heat,  290 

laughing  gas,  74 

Lavoisier,  70,  74,  76 

Le  Chatelier's  principle,  263 

Lenz's  law,  209 

lever,  15 

light,  60 

light,  velocity  of,  279 

lines  of  force,  208 


liquefaction  of  gases,  171,  222 
Lodge,  127 

machine,  8,  153 

magnetism,  32,  47,  204,  205 

man,  1 

manometer,  57 

Marconi,  124 

mass  action,  260 

mass  of  electron,  211 

mass  of  molecules,  277 

mathematical  physicist,  126 

mathematics,  115 

matter,  114,  303 

Maxwell,  124 

mean  free  path,  278 

measurement,  methods  of,  29 

mechanical  equivalent,  113,  289 

Mendelejeff,  100 

mercury  vapor  pump,  285 

meridian,  34 

meter,  122 

metric  units,  26,  123 

Michelson,  123 

milli-,  26 

Millikan,  276,  307,  316 

mina,  12 

mixtures,  method  of,  289 

mobility,  184,  255 

mole,  85 

molecular  conductivity,  250 

molecular  energy,  270 

molecular  impacts,  160 

molecular  mixtures,  234 

molecular  motions,  155 

molecules,  34,  71,  80,  82,  246,  291, 293 

molecules,  number  per  mole,  282 

momentum,  147,  148 

Moseley,  320 

motion,  64 

musical  note,  63 

mutual  energy,  electrical,  215 

natural  transformations,  267 
natural  vibrations,  298 
Newlands,  100 

Newton,  54,  79,  124,  137,  141,  148 
Newton's  laws,  139,  141,  147,  148 
neutralization  of  acid  and  base,  261 
niton,  313 
Nobel  prize,  123 
nucleus,  88,  96 


326 


INDEX 


null  method,  29 
numeric,  119 

objective  reality,  60 
Oersted,  205 
Ohm's  law,  194 
opposition  method,  29 
oscillators,  296 
osmotic  pressure,  237,  271 

TT,  44 

pace,  25 

paramagnetic,  204 

partial  pressure,  235 

Pascal,  54,  56 

Pascal's  law,  58 

pellate,  146 

pendulum,  32 

perfect  gas,  220,  222 

period,  33 

periodic  series,  91,  99 

perpetual  motion,  152 

Perrin,  271,  276 

Petit,  297 

phlogiston,  75,  77 

photoelectric,  315 

physics,  classification,  60 

physical  chemistry,  177 

pile,  Volta's,  175 

Pisa,  32,  135 

pitch,  61,  63 

Planck's  constant,  297,  315,  316 

Pliny,  49 

polonium,  92,  313 

pontifex,  49 

power,  145 

powers  of  ten,  71 

potential,  102,  106 

practice,  45 

pressure  coefficient  of  gas,  170 

pressure  of  gas,  156 

pressure  of  dissolved  gas,  235 

pressure  gauge,  287 

pressure  head,  136 

Priestley,  74,  76 

probability,  159 

prototype,  27 

Ptolemy,  50 

pulley,  19 

pulse,  62 

pyramids,  21 

Pythagoras,  28 


quantum,  295,  316 

R,  gas  constant,  218 

radial  energy,  196 

radiant  matter,  189 

radiation,  191 

radioactive,  92,  313 

radium,  97 

rarefaction,  pulse  of,  62 

rates,  128 

Raleigh,  270 

realities  of  science,  60 

recombinations  in  electrolysis,  250 

reflection,  301 

relative  motion,  64 

Renaissance,  51 

research  physicist,  126 

residual  rays,  300 

resonance,  300 

Roman  Empire,  48 

Romans,  22 

Rontgen,  93,  190 

roots  of  cubic,  223 

Rutherford,  321 

salt,  256 

Samothrace,   40 

saturated  vapor,  229 

scaffolding,  21 

scattering  of  alpha  rays,  320 

Scheele,  74,  76 

screw,  19 

second,  34 

self-induction,  coefficient  of,  214 

sexagesimal  system,  12 

shekel,  12 

Skeptical  Chemist,  74 

slavery,  22 

slope,  130 

solenoid,  202 

solute,  234 

solvent,  234 

sound,  60,  61 

span,  25 

spectral  lines,  319 

spectrum,  continuous,  186 

speed,  128,  129,  134 

statistical  method,  159 

stone,  27 

strain,  30 

stress,  30 

Stokes's  law,  273,  312 


INDEX 


327 


subjective  reality,  60 
substitution  method,  28 
supersaturated  vapor,  228 
surface  tension,  221 
symbolized  logic,  115 
sympathetic  vibrations,  300 
Syracuse,  44 

talent,  12 

tangential  energy,  196 

terrella,  47 

Thales,  13,  34 

theory,  34,  45 

thermionic  emission,  185 

thermodynamics,  laws  of,   114. 

268 

thermometer,  166,  170 
Thomson,  229,  230 
thorium,  92 
time,  32 
tool,  1,  4 

Torricelli,  54,  137 
Townsend,  306 
tractate,  146 
train  of  waves,  65 
transit,  34 

translation,  motion  of,  7 
triple  point,  241 

ultraviolet,  187 
ultimate  strength,  31 
United  States,  27 


263, 


units,  24,  123.     See  also  electricity, 

units  of 

universal  gravitation,  141,  142 
uranium,  92 

vacuum,  54,  90 

valence,  86 

Van't  Hoff,  239,  243 

Van  der  Waals's  equation,  222,  224, 

230,  281 

vapor  pressure,  225,  239,  240 
variable  mass,  213 
variable  standards,  25 
velocity,  134 
Vesuvius,  49 

vibration  of  elastic  system,  66 
viscosity,  274 
Vitruvius,  42 
volt,  178 
Volta,  86,  175,  197 

Waage,  259 

water,  decomposition  of,  253 

watt,  145 

wedge,  18 

weighing,  27 

weight,  147 

wheel  and  axle,  21 

wireless  telegraphy,  124 

work,  8.    See  energy 

work  principle,  20,  153 

X-rays,  91,  190 


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the  British  army.  In  the  form  of  fiction  and  with  vivid  de- 
scription and  illustration,  there  is  presented  the  scientific  inter- 
pretation of  life.  As  the  reflections  of  a  man  scientifically 
trained  who  has  been  for  four  years  in  the  presence  of  much 
suffering  and  death  and  can  still  believe  in  a  just  God  who  is 
not  only  the  First  Cause,  but  also  the  loving  Father  of  man- 
kind, the  volume  is  of  deep  significance  and  import.  The 
author's  style  is  attractive  and  often  eloquent.  What  he  has 
to  say  has  a  special  appeal  at  this  time  when  the  questions 
which  he  raises  are  being  brought  home  afresh  to  many  a  be- 
reaved family. 


Education  by  Violence 

BY   HENRY   S.    CANBY 

Cloth,  i2mo,  $1.50 

Professor  Canby  here  deals  with  the  effects  of  the  war  and 
the  rehabilitation  of  society  at  home  and  in  Europe.  Among 
the  specific  topics  taken  up  are  the  conditions  in  England  and 
France,  the  racial  and  spiritual  differences  and  agreements  be- 
tween the  Allies  and  the  prospects  of  a  peace  which  shall  finally 
end  war.  As  the  reflections  of  the  mind  of  an  American  scholar 
when  he  comes  into  contact  with  the  realities  of  the  war  and  its 
changes,  the  book  is  certain  to  have  a  wide  interest  in  this 
country. 


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TWO   NEW  BOOKS  ON   RECONSTRUCTION 


Reconstruction  and  National  Life 

BY   CECIL   FAIRFIELD    LAVELL 

Cloth ,  1 2 mo 

The  purpose  of  Professor  LavelPs  new  volume  is,  primarily,  to  sug- 
gest and  illustrate  an  historical  approach  to  the  problem  of  recon- 
struction in  Europe.  Professor  Lavell  will  be  remembered  as  author, 
with  Professor  Charles  E.  Payne,  of  "  Imperial  England,"  published 
in  the  fall  of  last  year. 

Problems  of  Reconstruction 

BY   ISAAC    LIPPINCOTT 

Associate  Professor  of  Economics,  Washington  University 

Cloth,  I2mo,  $1.60 

"  From  an  industrial  point  of  view  the  nations  at  war  are  con- 
fronted with  two  groups  of  problems.  Stated  briefly,  the  first  group 
contains  questions  of  concentrating  industrial  effort  largely  on  war 
production,  of  diverting  men,  materials  and  financial  resources  to  the 
essential  industries  and  of  curtailing  the  operations  of  all  the  rest,  of 
regulating  commerce  with  foreign  countries,  and  of  formulating 
policies  and  methods  for  the  accomplishment  of  these  ends.  In 
short,  this  is  principally  a  question  of  development  of  war  control 
with  all  that  this  implies.  The  second  group  of  problems  arises  out 
of  the  first.  It  involves  such  questions  as  the  dissolution  of  the  war 
organization,  the  removal  of  the  machinery  of  control,  the  restoration 
of  men,  funds,  and  materials  to  the  industries  which  serve  the  uses 
of  peace,  and  the  reestablishment  of  normal  commercial  relations 
with  the  outside  world.  The  latter  are  post-war  problems.  Their 
prompt  solution  is  necessary  because  the  war  has  turned  industrial 
and  social  life  into  new  channels,  and  because  it  will  be  necessary 
for  us  to  restore  the  normal  order  as  quickly  as  possible.  These 
brief  statements  outline  the  task  of  this  volume." 


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THE  WORLD'S  PEACE  PROBLEMS 

The  Great  Peace 

BY  H.  H.  POWERS 

Author  of  "  America  Among  the  Nations,"  "  The  Things  Men 
Fight  For,"  etc. 

Cloth,  i2mo,  $2.25 

"  The  terms  of  peace  to  be  agreed  upon  must  be  based  on  the  full- 
est recognition  of  the  special  problems  and  wishes  of  the  associated 
nations.  The  problem  of  problems  is  the  control  of  the  sea.  .  .  . 
These  questions  are  discussed  with  thoughtfulness  and  clarity,  and  a 
wide  grasp  of  circumstances  and  difficulties." —  The  Detroit  Free 
Press. 

National  Governments 
and  the  World  War 

BY  FREDERIC  A.  OGG 
Professor  of  Political  Science  in  the  University  of  Wisconsin 

AND 

CHARLES  A.  BEARD 
Director  of  the  Bureau  of  Municipal  Research,  New  York  City 

Cloth,  8°,  $2.50 

In  this  new  volume  Professors  Ogg  and  Beard  give  us  a  fuller 
realization  of  the  bearings  of  governmental  organization  and  practice 
upon  public  well-being,  a  better  knowledge  of  the  political  experience 
and  problems  of  other  peoples,  and  a  new  enthusiasm  for  national 
and  international  reconstruction  on  lines  such  as  will  conserve  the 
dearly  bought  gains  of  the  recent  conflict. 


The  End  of  the  War 


BY  WALTER  E.  WEYL 

Author  of  "  American  World  Policies,"  "  The  New  Democracy,"  etc. 

Cloth,  12°,  $2.00 

"Mr.  Weyl  says  sobering  and  important  things.  .  .  .  His  plea  is 
strong  and  clear  for  America  to  begin  to  establish  her  leadership  of 
the  democratic  forces  of  the  world  ...  to  insure  that  the  settlement 
of  the  war  is  made  on  lines  that  will  produce  international  amity 
everywhere." — N.  Y.  Times. 


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Tll'0  NEW  BOOKS  ON  CHINA 

China  and  the  World  War 

BY  W.  REGINALD  WHEELER 

Cloth,  ismo,  $1.75 

This  is  a  clear  and  succinct .  account  of  affairs  in  China 
since  the  outbreak  of  the  war. 

Among  the  events  which  the  author  discusses  are  the  re- 
lations of  Japan  and  China;  political  conditions  in  China 
with  the  conflict  between  republicanism  and  the  monarchical 
form  of  government;  the  American-Japanese  agreement, 
and  the  Chinese-Japanese  alliance  for  intervention  in  Siberia. 
Dr.  Wheeler  sets  these  matters  forth  in  a  simple,  straight- 
forward fashion  with  many  citations  from  Chinese  papers 
and  documents  which  provide  most  interesting  reading  for 
Westerners. 

The  author  is  a  professor  in  Hangchow  Christian  College, 
is  a  friend  of  the  Chinese  Republic,  and  writes  with  sym- 
pathy and  understanding  of  Chinese  affairs. 

Foreign  Financial  Control  in  China 

BY  T.  W.  OVERLACH 

Cloth,  12°,  $2.00 

This  is  a  careful  study  of  British,  Russian,  French,  Ger- 
man, Japanese,  and  American  financial  intervention  and 
financial  operations  in  China.  It  is  well  documented  and 
clearly  written.  It  treats  of  a  pressing  issue  in  the  Orient 
—  one  which  involves  the  United  States  deeply  and  will 
press  to  the  front  after  the  war.  It  is  the  first  consistent 
account  of  the  origin  and  development  of  foreign  control  in 
China.  In  view  of  our  interests  there,  both  practical  and 
political,  and  especially  on  account  of  the  present  crisis  and 
further  impending  crises,  there  is  significance  in  Mr.  Over- 
lach's  authoritative. 


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THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN     INITIAL     FINE     OF     25     CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  5O  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


NOV    12 

NOV  121932 

SEP  17  193.' 

FEB1 11998 


LD  21-50m-8,-32 


397609 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


